chapter 10 resource masters - north hunterdon-voorhees ... 11/chapter 11... ·...

Chapter 10Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 10 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828013-3 Algebra 2Chapter 10 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 10-1Study Guide and Intervention . . . . . . . . 573–574Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 575Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 576Reading to Learn Mathematics . . . . . . . . . . 577Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 578






Chapter 10 AssessmentChapter 10 Test, Form 1 . . . . . . . . . . . 609–610Chapter 10 Test, Form 2A . . . . . . . . . . 611–612Chapter 10 Test, Form 2B . . . . . . . . . . 613–614Chapter 10 Test, Form 2C . . . . . . . . . . 615–616Chapter 10 Test, Form 2D . . . . . . . . . . 617–618Chapter 10 Test, Form 3 . . . . . . . . . . . 619–620Chapter 10 Open-Ended Assessment . . . . . 621Chapter 10 Vocabulary Test/Review . . . . . . 622Chapter 10 Quizzes 1 & 2 . . . . . . . . . . . . . . 623Chapter 10 Quizzes 3 & 4 . . . . . . . . . . . . . . 624Chapter 10 Mid-Chapter Test . . . . . . . . . . . . 625Chapter 10 Cumulative Review . . . . . . . . . . 626Chapter 10 Standardized Test Practice . 627–628Unit 3 Test/Review (Ch. 8–10) . . . . . . . 629–630

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A30

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 10 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 10 Resource Masters includes the core materialsneeded for Chapter 10. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 10-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 10Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 572–573. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

1010

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 10.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

Change of Base Formula

common logarithm

LAW·guh·RIH·thuhm

exponential decay

EHK·spuh·NEHN·chuhl

exponential equation

exponential function

exponential growth

exponential inequality

(continued on the next page)

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

logarithm

logarithmic function

LAW·guh·RIHTH·mihk

natural base, e

natural base exponential function

natural logarithm

natural logarithmic function

rate of decay

rate of growth

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

1010

Study Guide and InterventionExponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

© Glencoe/McGraw-Hill 573 Glencoe Algebra 2

Less

on

10-

1

Exponential Functions An exponential function has the form y � abx,where a � 0, b � 0, and b � 1.

1. The function is continuous and one-to-one.

Properties of an2. The domain is the set of all real numbers.

Exponential Function3. The x-axis is the asymptote of the graph.4. The range is the set of all positive numbers if a � 0 and all negative numbers if a � 0.5. The graph contains the point (0, a).

Exponential Growth If a � 0 and b � 1, the function y � abx represents exponential growth.and Decay If a � 0 and 0 � b � 1, the function y � abx represents exponential decay.

Sketch the graph of y � 0.1(4)x. Then state the function’s domain and range.Make a table of values. Connect the points to form a smooth curve.

The domain of the function is all real numbers, while the range is the set of all positive real numbers.

Determine whether each function represents exponential growth or decay.a. y � 0.5(2)x b. y � �2.8(2)x c. y � 1.1(0.5)x

exponential growth, neither, since �2.8, exponential decay, sincesince the base, 2, is the value of a is less the base, 0.5, is betweengreater than 1 than 0. 0 and 1

Sketch the graph of each function. Then state the function’s domain and range.

1. y � 3(2)x 2. y � �2� �x

3. y � 0.25(5)x

Domain: all real Domain: all real Domain: all real numbers; Range: all numbers; Range: all numbers; Range: allpositive real numbers negative real numbers positive real numbers

Determine whether each function represents exponential growth or decay.

4. y � 0.3(1.2)x growth 5. y � �5� �x

neither 6. y � 3(10)�x decay4�5

x

y

O

x

y

O

x

y

O

1�4

x �1 0 1 2 3

y 0.025 0.1 0.4 1.6 6.4

x

y

O

Example 1Example 1

Example 2Example 2

ExercisesExercises


Exponential Equations and Inequalities All the properties of rational exponentsthat you know also apply to real exponents. Remember that am � an � am � n, (am)n � amn,and am an � am � n.

Property of Equality for If b is a positive number other than 1,Exponential Functions then bx � by if and only if x � y.

Property of Inequality forIf b � 1

Exponential Functionsthen bx � by if and only if x � yand bx � by if and only if x � y.

Study Guide and Intervention (continued)

Exponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

Solve 4x � 1 � 2x � 5.4x � 1 � 2x � 5 Original equation

(22)x � 1 � 2x � 5 Rewrite 4 as 22.

2(x � 1) � x � 5 Prop. of Inequality for ExponentialFunctions

2x � 2 � x � 5 Distributive Property

x � 7 Subtract x and add 2 to each side.

Solve 52x � 1 � .

52x � 1 � Original inequality

52x � 1 � 5�3 Rewrite as 5�3.

2x � 1 � �3 Prop. of Inequality for Exponential Functions

2x � �2 Add 1 to each side.

x � �1 Divide each side by 2.

The solution set is {x|x � �1}.

1�125

1�125

1�125

Example 1Example 1 Example 2Example 2

ExercisesExercises

Simplify each expression.

1. (3�2�)�2� 2. 25�2� � 125�2� 3. (x�2�y3�2�)�2�

9 55�2� or 3125�2� x2y6

4. (x�6�)(x�5�) 5. (x�6�)�5� 6. (2x)(5x3)x�6� � �5� x�30� 10x4�

Solve each equation or inequality. Check your solution.

7. 32x � 1 � 3x � 2 3 8. 23x � 4x � 2 4 9. 32x � 1 � �

10. 4x � 1 � 82x � 3 � 11. 8x � 2 � 12. 252x � 125x � 2 6

13. 4�x� � 16�5� 20 14. x�3� � 36��34�

6 15. x�2� � 81��18�

�

3

16. 3x � 4 � x � 1 17. 42x � 2 � 2x � 1 x � 18. 52x � 125x � 5 x � 15

19. 104x � 1 � 100x � 2 20. 73x � 49x2 21. 82x � 5 � 4x � 8

x � � x � or x � 0 x � �341�

3�2

5�2

5�3

1�27

2�3

1�16

7�4

1�2

1�9

Skills PracticeExponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1


Less

on

10-

1


1. y � 3(2)x 2. y � 2� �x

domain: all real numbers; domain: all real numbers;range: all positive numbers range: all positive numbers


3. y � 3(6)x growth 4. y � 2� �xdecay

5. y � 10�x decay 6. y � 2(2.5)x growth

Write an exponential function whose graph passes through the given points.

7. (0, 1) and (�1, 3) y � � �x8. (0, 4) and (1, 12) y � 4(3)x

9. (0, 3) and (�1, 6) y � 3� �x10. (0, 5) and (1, 15) y � 5(3)x

11. (0, 0.1) and (1, 0.5) y � 0.1(5)x 12. (0, 0.2) and (1, 1.6) y � 0.2(8)x


13. (3�3�)�3� 27 14. (x�2�)�7� x�14�

15. 52�3� � 54�3� 56�3� 16. x3 x x2�


17. 3x � 9 x � 2 18. 22x � 3 � 32 1

19. 49x � x � � 20. 43x � 2 � 16

21. 32x � 5 � 27x 5 22. 27x � 32x � 3 3

4�3

1�2

1�7

1�2

1�3

9�10

x

y

Ox

y

O

1�2



1. y � 1.5(2)x 2. y � 4(3)x 3. y � 3(0.5)x

domain: all real domain: all real domain: all real numbers; range: all numbers; range: all numbers; range: all positive numbers positive numbers positive numbers


4. y � 5(0.6)x decay 5. y � 0.1(2)x growth 6. y � 5 � 4�x decay

Write an exponential function whose graph passes through the given points.

7. (0, 1) and (�1, 4) 8. (0, 2) and (1, 10) 9. (0, �3) and (1, �1.5)

y � � �xy � 2(5)x y � �3(0.5)x

10. (0, 0.8) and (1, 1.6) 11. (0, �0.4) and (2, �10) 12. (0, ) and (3, 8)

y � 0.8(2)x y � �0.4(5)x y � �(2)x


13. (2�2�)�8� 16 14. (n�3�)�75� n15 15. y�6� � y5�6� y6�6�

16. 13�6� � 13�24� 133�6� 17. n3 n n3 � � 18. 125�11� 5�11� 52�11�


19. 33x � 5 � 81 x � 3 20. 76x � 72x � 20 �5 21. 36n � 5 � 94n � 3 n �

22. 92x � 1 � 27x � 4 14 23. 23n � 1 � � �nn 24. 164n � 1 � 1282n � 1

BIOLOGY For Exercises 25 and 26, use the following information.The initial number of bacteria in a culture is 12,000. The number after 3 days is 96,000.

25. Write an exponential function to model the population y of bacteria after x days.y � 12,000(2)x

26. How many bacteria are there after 6 days? 768,000

27. EDUCATION A college with a graduating class of 4000 students in the year 2002predicts that it will have a graduating class of 4862 in 4 years. Write an exponentialfunction to model the number of students y in the graduating class t years after 2002.y � 4000(1.05)t

11�2

1�6

1�8

1�2

1�4

x

y

Ox

y

O

Practice (Average)

Exponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

Reading to Learn MathematicsExponential Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1


Less

on

10-

1

Pre-Activity How does an exponential function describe tournament play?

Read the introduction to Lesson 10-1 at the top of page 523 in your textbook.

How many rounds of play would be needed for a tournament with 100players? 7

Reading the Lesson

1. Indicate whether each of the following statements about the exponential function y � 10x is true or false.

a. The domain is the set of all positive real numbers. false

b. The y-intercept is 1. true

c. The function is one-to-one. true

d. The y-axis is an asymptote of the graph. false

e. The range is the set of all real numbers. false

2. Determine whether each function represents exponential growth or decay.

a. y � 0.2(3)x. growth b. y � 3� �x. decay c. y � 0.4(1.01)x. growth

3. Supply the reason for each step in the following solution of an exponential equation.

92x � 1 � 27x Original equation

(32)2x � 1 � (33)x Rewrite each side with a base of 3.32(2x � 1) � 33x Power of a Power

2(2x � 1) � 3x Property of Equality for Exponential Functions4x � 2 � 3x Distributive Propertyx � 2 � 0 Subtract 3x from each side.

x � 2 Add 2 to each side.

Helping You Remember

4. One way to remember that polynomial functions and exponential functions are differentis to contrast the polynomial function y � x2 and the exponential function y � 2x. Tell atleast three ways they are different.

Sample answer: In y � x2, the variable x is a base, but in y � 2x, thevariable x is an exponent. The graph of y � x2 is symmetric with respectto the y-axis, but the graph of y � 2x is not. The graph of y � x2 touchesthe x-axis at (0, 0), but the graph of y � 2x has the x-axis as an asymptote.You can compute the value of y � x2 mentally for x � 100, but you cannotcompute the value of y � 2x mentally for x � 100.

2�5


Finding Solutions of xy � yx

Perhaps you have noticed that if x and y are interchanged in equations suchas x � y and xy � 1, the resulting equation is equivalent to the originalequation. The same is true of the equation xy � yx. However, findingsolutions of xy � yx and drawing its graph is not a simple process.

Solve each problem. Assume that x and y are positive real numbers.

1. If a � 0, will (a, a) be a solution of xy � yx? Justify your answer.

2. If c � 0, d � 0, and (c, d) is a solution of xy � yx, will (d, c) also be a solution? Justify your answer.

3. Use 2 as a value for y in xy � yx. The equation becomes x2 � 2x.

a. Find equations for two functions, f(x) and g(x) that you could graph tofind the solutions of x2 � 2x. Then graph the functions on a separatesheet of graph paper.

b. Use the graph you drew for part a to state two solutions for x2 � 2x.Then use these solutions to state two solutions for xy � yx.

4. In this exercise, a graphing calculator will be very helpful. Use the technique of Exercise 3 to complete the tables below. Then graph xy � yx

for positive values of x and y. If there are asymptotes, show them in yourdiagram using dotted lines. Note that in the table, some values of y callfor one value of x, others call for two.

x

y

O

x y

4

4

5

5

8

8

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

x y

�12

�

�34

�

1

2

2

3

3

Study Guide and InterventionLogarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2


Less

on

10-

2

Logarithmic Functions and Expressions

Definition of Logarithm Let b and x be positive numbers, b � 1. The logarithm of x with base b is denoted with Base b logb x and is defined as the exponent y that makes the equation by � x true.

The inverse of the exponential function y � bx is the logarithmic function x � by.This function is usually written as y � logb x.

1. The function is continuous and one-to-one.

Properties of2. The domain is the set of all positive real numbers.

Logarithmic Functions3. The y-axis is an asymptote of the graph.4. The range is the set of all real numbers.5. The graph contains the point (0, 1).

Write an exponential equation equivalent to log3 243 � 5.35 � 243

Write a logarithmic equation equivalent to 6�3 � .

log6 � �3

Evaluate log8 16.

8�43

�

� 16, so log8 16 � .

Write each equation in logarithmic form.

1. 27 � 128 2. 3�4 � 3. � �3�

log2 128 � 7 log3 � �4 log�17

� � 3

Write each equation in exponential form.

4. log15 225 � 2 5. log3 � �3 6. log4 32 �

152 � 225 3�3 � 4�52

�� 32

Evaluate each expression.

7. log4 64 3 8. log2 64 6 9. log100 100,000 2.5

10. log5 625 4 11. log27 81 12. log25 5

13. log2 �7 14. log10 0.00001 �5 15. log4 �2.51�32

1�128

1�2

4�3

1�27

5�2

1�27

1�343

1�81

1�343

1�7

1�81

4�3

1�216

1�216

Example 1Example 1

Example 2Example 2

Example 3Example 3

ExercisesExercises


Solve Logarithmic Equations and Inequalities

Logarithmic to If b � 1, x � 0, and logb x � y, then x � by.Exponential Inequality If b � 1, x � 0, and logb x � y, then 0 � x � by.

Property of Equality for If b is a positive number other than 1, Logarithmic Functions then logb x � logb y if and only if x � y.

Property of Inequality for If b � 1, then logb x � logb y if and only if x � y, Logarithmic Functions and logb x � logb y if and only if x � y.


Logarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2

Solve log2 2x � 3.log2 2x � 3 Original equation

2x � 23 Definition of logarithm

2x � 8 Simplify.

x � 4 Simplify.

The solution is x � 4.

Solve log5 (4x � 3) � 3.log5 (4x � 3) � 3 Original equation

0 � 4x � 3 � 53 Logarithmic to exponential inequality

3 � 4x � 125 � 3 Addition Property of Inequalities

� x � 32 Simplify.

The solution set is �x | � x � 32�.3�4

3�4


ExercisesExercises

Solve each equation or inequality.

1. log2 32 � 3x 2. log3 2c � �2

3. log2x 16 � �2 4. log25 � � � 10

5. log4 (5x � 1) � 2 3 6. log8 (x � 5) � 9

7. log4 (3x � 1) � log4 (2x � 3) 4 8. log2 (x2 � 6) � log2 (2x � 2) 4

9. logx � 4 27 � 3 �1 10. log2 (x �3) � 4 13

11. logx 1000 � 3 10 12. log8 (4x � 4) � 2 15

13. log2 2x � 2 x � 2 14. log5 x � 2 x � 25

15. log2 (3x � 1) � 4 � � x � 5 16. log4 (2x) � � x �

17. log3 (x � 3) � 3 �3 � x � 24 18. log27 6x � x �3�2

2�3

1�4

1�2

1�3

2�3

1�2

x�2

1�8

1�18

5�3

Skills PracticeLogarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2


Less

on

10-

2


1. 23 � 8 log2 8 � 3 2. 32 � 9 log3 9 � 2

3. 8�2 � log8 � �2 4. � �2� log�

13

� � 2


5. log3 243 � 5 35 � 243 6. log4 64 � 3 43 � 64

7. log9 3 � 9�12

�� 3 8. log5 � �2 5�2 �


9. log5 25 2 10. log9 3

11. log10 1000 3 12. log125 5

13. log4 �3 14. log5 �4

15. log8 83 3 16. log27 �

Solve each equation or inequality. Check your solutions.

17. log3 x � 5 243 18. log2 x � 3 8

19. log4 y � 0 0 � y � 1 20. log�14

� x � 3

21. log2 n � �2 n � 22. logb 3 � 9

23. log6 (4x � 12) � 2 6 24. log2 (4x � 4) � 5 x � 9

25. log3 (x � 2) � log3 (3x) 1 26. log6 (3y � 5) � log6 (2y � 3) y 8

1�2

1�4

1�64

1�3

1�3

1�625

1�64

1�3

1�2

1�25

1�25

1�2

1�9

1�9

1�3

1�64

1�64



1. 53 � 125 log5 125 � 3 2. 70 � 1 log7 1 � 0 3. 34 � 81 log3 81 � 4

4. 3�4 � 5. � �3� 6. 7776

�15

�

� 6

log3 � �4 log�14

� � 3 log7776 6 �


7. log6 216 � 3 63 � 216 8. log2 64 � 6 26 � 64 9. log3 � �4 3�4 �

10. log10 0.00001 � �5 11. log25 5 � 12. log32 8 �

10�5 � 0.00001 25�12

�� 5 32

�35

�� 8


13. log3 81 4 14. log10 0.0001 �4 15. log2 �4 16. log�13

� 27 �3

17. log9 1 0 18. log8 4 19. log7 �2 20. log6 64 4

21. log3 �1 22. log4 �4 23. log9 9(n � 1) n � 1 24. 2log2 32 32

Solve each equation or inequality. Check your solutions.

25. log10 n � �3 26. log4 x � 3 x � 64 27. log4 x � 8

28. log�15

� x � �3 125 29. log7 q � 0 0 � q � 1 30. log6 (2y � 8) � 2 y 14

31. logy 16 � �4 32. logn � �3 2 33. logb 1024 � 5 4

34. log8 (3x � 7) � log8 (7x � 4) 35. log7 (8x � 20) � log7 (x � 6) 36. log3 (x2 � 2) � log3 x

x � �2 2

37. SOUND Sounds that reach levels of 130 decibels or more are painful to humans. Whatis the relative intensity of 130 decibels? 1013

38. INVESTING Maria invests $1000 in a savings account that pays 8% interestcompounded annually. The value of the account A at the end of five years can bedetermined from the equation log A � log[1000(1 � 0.08)5]. Find the value of A to thenearest dollar. $1469

3�4

1�8

1�2

3�2

1�1000

1�256

1�3

1�49

2�3

1�16

3�5

1�2

1�81

1�81

1�5

1�64

1�81

1�64

1�4

1�81

Practice (Average)

Logarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2

Reading to Learn MathematicsLogarithms and Logarithmic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2


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Pre-Activity Why is a logarithmic scale used to measure sound?


How many times louder than a whisper is normal conversation?104 or 10,000 times

Reading the Lesson1. a. Write an exponential equation that is equivalent to log3 81 � 4. 34 � 81

b. Write a logarithmic equation that is equivalent to 25��12

�� . log25 � �

c. Write an exponential equation that is equivalent to log4 1 � 0. 40 � 1

d. Write a logarithmic equation that is equivalent to 10�3 � 0.001. log10 0.001 � �3

e. What is the inverse of the function y � 5x? y � log5 x

f. What is the inverse of the function y � log10 x? y � 10x

2. Match each function with its graph.

a. y � 3x IV b. y � log3 x I c. y � � �xII

I. II. III.

3. Indicate whether each of the following statements about the exponential function y � log5 x is true or false.

a. The y-axis is an asymptote of the graph. trueb. The domain is the set of all real numbers. falsec. The graph contains the point (5, 0). falsed. The range is the set of all real numbers. truee. The y-intercept is 1. false

Helping You Remember4. An important skill needed for working with logarithms is changing an equation between

logarithmic and exponential forms. Using the words base, exponent, and logarithm, describean easy way to remember and apply the part of the definition of logarithm that says,“logb x � y if and only if by � x.” Sample answer: In these equations, b standsfor base. In log form, b is the subscript, and in exponential form, b is thenumber that is raised to a power. A logarithm is an exponent, so y, which isthe log in the first equation, becomes the exponent in the second equation.

x

y

Ox

y

O

x

y

O

1�3

1�2

1�5

1�5


Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

10-210-2

Musical RelationshipsThe frequencies of notes in a musical scale that are one octave apart arerelated by an exponential equation. For the eight C notes on a piano, theequation is Cn � C12n � 1, where Cn represents the frequency of note Cn.

1. Find the relationship between C1 and C2.

2. Find the relationship between C1 and C4.

The frequencies of consecutive notes are related by a common ratio r. The general equation is fn � f1rn � 1.

3. If the frequency of middle C is 261.6 cycles per second and the frequency of the next higher C is 523.2 cycles per second, find the common ratio r. (Hint: The two C’s are 12 notes apart.) Write the answer as a radicalexpression.

4. Substitute decimal values for r and f1 to find a specific equation for fn.

5. Find the frequency of F# above middle C.

6. Frets are a series of ridges placed across the fingerboard of a guitar. Theyare spaced so that the sound made by pressing a string against one frethas about 1.0595 times the wavelength of the sound made by using thenext fret. The general equation is wn � w0(1.0595)n. Describe thearrangement of the frets on a guitar.

Study Guide and InterventionProperties of Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-310-3


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Properties of Logarithms Properties of exponents can be used to develop thefollowing properties of logarithms.

Product Property For all positive numbers m, n, and b, where b � 1, of Logarithms logb mn � logb m � logb n.

Quotient Property For all positive numbers m, n, and b, where b � 1, of Logarithms logb �

mn

� � logb m � logb n.

Power Property For any real number p and positive numbers m and b, of Logarithms where b � 1, logb mp � p logb m.

Use log3 28 � 3.0331 and log3 4 � 1.2619 to approximate the value of each expression.

ExampleExample

a. log3 36

log3 36 � log3 (32 � 4)� log3 32 � log3 4� 2 � log3 4� 2 � 1.2619� 3.2619

b. log3 7

log3 7 � log3 � �� log3 28 � log3 4� 3.0331 � 1.2619� 1.7712

c. log3 256

log3 256 � log3 (44)� 4 � log3 4� 4(1.2619)� 5.0476

28�4

ExercisesExercises

Use log12 3 � 0.4421 and log12 7 � 0.7831 to evaluate each expression.

1. log12 21 1.2252 2. log12 0.3410 3. log12 49 1.5662

4. log12 36 1.4421 5. log12 63 1.6673 6. log12 �0.2399

7. log12 0.2022 8. log12 16,807 3.9155 9. log12 441 2.4504

Use log5 3 � 0.6826 and log5 4 � 0.8614 to evaluate each expression.

10. log5 12 1.5440 11. log5 100 2.8614 12. log5 0.75 �0.1788

13. log5 144 3.0880 14. log5 0.3250 15. log5 375 3.6826

16. log5 1.3� 0.1788 17. log5 �0.3576 18. log5 1.730481�5

9�16

27�16

81�49

27�49

7�3


Solve Logarithmic Equations You can use the properties of logarithms to solveequations involving logarithms.

Solve each equation.

a. 2 log3 x � log3 4 � log3 25

2 log3 x � log3 4 � log3 25 Original equation

log3 x2 � log3 4 � log3 25 Power Property

log3 � log3 25 Quotient Property

� 25 Property of Equality for Logarithmic Functions

x2 � 100 Multiply each side by 4.

x � �10 Take the square root of each side.

Since logarithms are undefined for x � 0, �10 is an extraneous solution.The only solution is 10.

b. log2 x � log2 (x � 2) � 3

log2 x � log2 (x � 2) � 3 Original equation

log2 x(x � 2) � 3 Product Property

x(x � 2) � 23 Definition of logarithm

x2 � 2x � 8 Distributive Property

x2 � 2x � 8 � 0 Subtract 8 from each side.

(x � 4)(x � 2) � 0 Factor.

x � 2 or x � �4 Zero Product Property

Since logarithms are undefined for x � 0, �4 is an extraneous solution.The only solution is 2.

Solve each equation. Check your solutions.

1. log5 4 � log5 2x � log5 24 3 2. 3 log4 6 � log4 8 � log4 x 27

3. log6 25 � log6 x � log6 20 4 4. log2 4 � log2 (x � 3) � log2 8 �

5. log6 2x � log6 3 � log6 (x � 1) 3 6. 2 log4 (x � 1) � log4 (11 � x) 2

7. log2 x � 3 log2 5 � 2 log2 10 12,500 8. 3 log2 x � 2 log2 5x � 2 100

9. log3 (c � 3) � log3 (4c � 1) � log3 5 10. log5 (x � 3) � log5 (2x � 1) � 24�7

8�19

5�2

1�2

x2�4

x2�4


Properties of Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

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ExampleExample

ExercisesExercises

Skills PracticeProperties of Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

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Use log2 3 � 1.5850 and log2 5 � 2.3219 to approximate the value of eachexpression.

1. log2 25 4.6438 2. log2 27 4.755

3. log2 �0.7369 4. log2 0.7369

5. log2 15 3.9069 6. log2 45 5.4919

7. log2 75 6.2288 8. log2 0.6 �0.7369

9. log2 �1.5850 10. log2 0.8481


11. log10 27 � 3 log10 x 3 12. 3 log7 4 � 2 log7 b 8

13. log4 5 � log4 x � log4 60 12 14. log6 2c � log6 8 � log6 80 5

15. log5 y � log5 8 � log5 1 8 16. log2 q � log2 3 � log2 7 21

17. log9 4 � 2 log9 5 � log9 w 100 18. 3 log8 2 � log8 4 � log8 b 2

19. log10 x � log10 (3x � 5) � log10 2 2 20. log4 x � log4 (2x � 3) � log4 2 2

21. log3 d � log3 3 � 3 9 22. log10 y � log10 (2 � y) � 0 1

23. log2 s � 2 log2 5 � 0 24. log2 (x � 4) � log2 (x � 3) � 3 4

25. log4 (n � 1) � log4 (n � 2) � 1 3 26. log5 10 � log5 12 � 3 log5 2 � log5 a 15

1�25

9�5

1�3

5�3

3�5


Use log10 5 � 0.6990 and log10 7 � 0.8451 to approximate the value of eachexpression.

1. log10 35 1.5441 2. log10 25 1.3980 3. log10 0.1461 4. log10 �0.1461

5. log10 245 2.3892 6. log10 175 2.2431 7. log10 0.2 �0.6990 8. log10 0.5529


9. log7 n � log7 8 4 10. log10 u � log10 4 8

11. log6 x � log6 9 � log6 54 6 12. log8 48 � log8 w � log8 4 12

13. log9 (3u � 14) � log9 5 � log9 2u 2 14. 4 log2 x � log2 5 � log2 405 3

15. log3 y � �log3 16 � log3 64 16. log2 d � 5 log2 2 � log2 8 4

17. log10 (3m � 5) � log10 m � log10 2 2 18. log10 (b � 3) � log10 b � log10 4 1

19. log8 (t � 10) � log8 (t � 1) � log8 12 2 20. log3 (a � 3) � log3 (a � 2) � log3 6 0

21. log10 (r � 4) � log10 r � log10 (r � 1) 2 22. log4 (x2 � 4) � log4 (x � 2) � log4 1 3

23. log10 4 � log10 w � 2 25 24. log8 (n � 3) � log8 (n � 4) � 1 4

25. 3 log5 (x2 � 9) � 6 � 0 �4 26. log16 (9x � 5) � log16 (x2 � 1) � 3

27. log6 (2x � 5) � 1 � log6 (7x � 10) 8 28. log2 (5y � 2) � 1 � log2 (1 � 2y) 0

29. log10 (c2 � 1) � 2 � log10 (c � 1) 101 30. log7 x � 2 log7 x � log7 3 � log7 72 6

31. SOUND The loudness L of a sound in decibels is given by L � 10 log10 R, where R is thesound’s relative intensity. If the intensity of a certain sound is tripled, by how manydecibels does the sound increase? about 4.8 db

32. EARTHQUAKES An earthquake rated at 3.5 on the Richter scale is felt by many people,and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitudereading m is given by m � log10 x, where x represents the amplitude of the seismic wavecausing ground motion. How many times greater is the amplitude of an earthquake thatmeasures 4.5 on the Richter scale than one that measures 3.5? 10 times

1�2

1�4

1�3

3�2

2�3

25�7

5�7

7�5

Practice (Average)

Properties of Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-310-3

Reading to Learn MathematicsProperties of Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-310-3


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Pre-Activity How are the properties of exponents and logarithms related?


Find the value of log5 125. 3 Find the value of log5 5. 1Find the value of log5 (125 � 5). 2Which of the following statements is true? BA. log5 (125 � 5) � (log5 125) � (log5 5)

B. log5 (125 � 5) � log5 125 � log5 5

Reading the Lesson1. Each of the properties of logarithms can be stated in words or in symbols. Complete the

statements of these properties in words.

a. The logarithm of a quotient is the of the logarithms of the

and the .

b. The logarithm of a power is the of the logarithm of the base and

the .

c. The logarithm of a product is the of the logarithms of its

.

2. State whether each of the following equations is true or false. If the statement is true,name the property of logarithms that is illustrated.

a. log3 10 � log3 30 � log3 3 true; Quotient Propertyb. log4 12 � log4 4 � log4 8 falsec. log2 81 � 2 log2 9 true; Power Propertyd. log8 30 � log8 5 � log8 6 false

3. The algebraic process of solving the equation log2 x � log2 (x � 2) � 3 leads to “x � �4or x � 2.” Does this mean that both �4 and 2 are solutions of the logarithmic equation?Explain your reasoning. Sample answer: No; 2 is a solution because it checks: log2 2 � log2 (2 � 2) � log2 2 � log2 4 � 1 � 2 � 3. However,because log2 (�4) and log2 (� 2) are undefined, �4 is an extraneoussolution and must be eliminated. The only solution is 2.

Helping You Remember4. A good way to remember something is to relate it something you already know. Use words

to explain how the Product Property for exponents can help you remember the productproperty for logarithms. Sample answer: When you multiply two numbers orexpressions with the same base, you add the exponents and keep thesame base. Logarithms are exponents, so to find the logarithm of aproduct, you add the logarithms of the factors, keeping the same base.

factorssum

exponentproduct

denominatornumeratordifference


SpiralsConsider an angle in standard position with its vertex at a point O called thepole. Its initial side is on a coordinatized axis called the polar axis. A point Pon the terminal side of the angle is named by the polar coordinates (r, �),where r is the directed distance of the point from O and � is the measure ofthe angle. Graphs in this system may be drawn on polar coordinate papersuch as the kind shown below.

1. Use a calculator to complete the table for log2r � �12�0�.

(Hint: To find � on a calculator, press 120 r 2 .)

2. Plot the points found in Exercise 1 on the grid above and connect to form a smooth curve.

This type of spiral is called a logarithmic spiral because the angle measures are proportional to the logarithms of the radii.

r 1 2 3 4 5 6 7 8

) LOG�) LOG�

0

10

20

30

40

5060

708090100

110120

130

140

150

160

170

180

190

200

210

220

230

240250

260 270 280290

300310

320

330

340

350

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionCommon Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

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Common Logarithms Base 10 logarithms are called common logarithms. Theexpression log10 x is usually written without the subscript as log x. Use the key onyour calculator to evaluate common logarithms.The relation between exponents and logarithms gives the following identity.

Inverse Property of Logarithms and Exponents 10log x � x

Evaluate log 50 to four decimal places.Use the LOG key on your calculator. To four decimal places, log 50 � 1.6990.

Solve 32x � 1 � 12.32x � 1 � 12 Original equation

log 32x � 1 � log 12 Property of Equality for Logarithms

(2x � 1) log 3 � log 12 Power Property of Logarithms

2x � 1 � Divide each side by log 3.

2x � � 1 Subtract 1 from each side.

x � � � 1� Multiply each side by .

x � 0.6309

Use a calculator to evaluate each expression to four decimal places.

1. log 18 2. log 39 3. log 1201.2553 1.5911 2.0792

4. log 5.8 5. log 42.3 6. log 0.0030.7634 1.6263 �2.5229

Solve each equation or inequality. Round to four decimal places.

7. 43x � 12 0.5975 8. 6x � 2 � 18 �0.3869

9. 54x � 2 � 120 1.2437 10. 73x � 1 � 21 {x |x � 0.8549}

11. 2.4x � 4 � 30 �0.1150 12. 6.52x � 200 {x |x � 1.4153}

13. 3.64x � 1 � 85.4 1.1180 14. 2x � 5 � 3x � 2 13.9666

15. 93x � 45x � 2 �8.1595 16. 6x � 5 � 27x � 3 �3.6069

1�2

log 12�log 3

1�2

log 12�log 3

log 12�log 3

LOG

ExercisesExercises

Example 1Example 1

Example 2Example 2


Change of Base Formula The following formula is used to change expressions withdifferent logarithmic bases to common logarithm expressions.

Change of Base Formula For all positive numbers a, b, and n, where a � 1 and b � 1, loga n �

Express log8 15 in terms of common logarithms. Then approximateits value to four decimal places.

log8 15 � Change of Base Formula

� 1.3023 Simplify.

The value of log8 15 is approximately 1.3023.

Express each logarithm in terms of common logarithms. Then approximate itsvalue to four decimal places.

1. log3 16 2. log2 40 3. log5 35

, 2.5237 , 5.3219 , 2.2091

4. log4 22 5. log12 200 6. log2 50

, 2.2297 , 2.1322 , 5.6439

7. log5 0.4 8. log3 2 9. log4 28.5

, �0.5693 , 0.6309 , 2.4164

10. log3 (20)2 11. log6 (5)4 12. log8 (4)5

, 5.4537 , 3.5930 , 3.3333

13. log5 (8)3 14. log2 (3.6)6 15. log12 (10.5)4

, 3.8761 , 11.0880 , 3.7851

16. log3 �150� 17. log43�39� 18. log5

4�1600�

, 2.2804 , 0.8809 , 1.1460log 1600��4 log 5

log 39�3 log 4

log 150�2 log 3

4 log 10.5��

log 126 log 3.6��

log 23 log 8�log 5

5 log 4�log 8

4 log 5�log 6

2 log 20��

log 3

log 28.5��

log 4log 2�log 3

log 0.4�log 5

log 50�log 2

log 200�log 12

log 22�log 4

log 35�log 5

log 40�log 2

log 16�log 3

log10 15�log10 8

logb n�logb a


Common Logarithms

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ExampleExample

ExercisesExercises

Skills PracticeCommon Logarithms

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1. log 6 0.7782 2. log 15 1.1761

3. log 1.1 0.0414 4. log 0.3 �0.5229

Use the formula pH � �log[H�] to find the pH of each substance given itsconcentration of hydrogen ions.

5. gastric juices: [H�] � 1.0 � 10�1 mole per liter 1.0

6. tomato juice: [H�] � 7.94 � 10�5 mole per liter 4.1

7. blood: [H�] � 3.98 � 10�8 mole per liter 7.4

8. toothpaste: [H�] � 1.26 � 10�10 mole per liter 9.9


9. 3x � 243 {x |x � 5} 10. 16v �v �v � � �11. 8p � 50 1.8813 12. 7y � 15 1.3917

13. 53b � 106 0.9659 14. 45k � 37 0.5209

15. 127p � 120 0.2752 16. 92m � 27 0.75

17. 3r � 5 � 4.1 6.2843 18. 8y � 4 � 15 {y |y � �2.6977}

19. 7.6d � 3 � 57.2 �1.0048 20. 0.5t � 8 � 16.3 3.9732

21. 42x2� 84 �1.0888 22. 5x2 � 1� 10 �0.6563


23. log3 7 ; 1.7712 24. log5 66 ; 2.6032

25. log2 35 ; 5.1293 26. log6 10 ; 1.2851log10 10��log10 6

log10 35��log10 2

log10 66��log10 5

log10 7�log10 3

1�2

1�4



1. log 101 2.0043 2. log 2.2 0.3424 3. log 0.05 �1.3010

Use the formula pH � �log[H�] to find the pH of each substance given itsconcentration of hydrogen ions.

4. milk: [H�] � 2.51 � 10�7 mole per liter 6.6

5. acid rain: [H�] � 2.51 � 10�6 mole per liter 5.6

6. black coffee: [H�] � 1.0 � 10�5 mole per liter 5.0

7. milk of magnesia: [H�] � 3.16 � 10�11 mole per liter 10.5


8. 2x 25 {x |x 4.6439} 9. 5a � 120 2.9746 10. 6z � 45.6 2.1319

11. 9m � 100 {m |m � 2.0959} 12. 3.5x � 47.9 3.0885 13. 8.2y � 64.5 1.9802

14. 2b � 1 7.31 {b |b � 1.8699} 15. 42x � 27 1.1887 16. 2a � 4 � 82.1 10.3593

17. 9z � 2 � 38 {z |z � 3.6555} 18. 5w � 3 � 17 �1.2396 19. 30x2� 50 �1.0725

20. 5x2 � 3 � 72 �2.3785 21. 42x � 9x � 1 3.8188 22. 2n � 1 � 52n � 1 0.9117


23. log5 12 ; 1.5440 24. log8 32 ; 1.6667 25. log11 9 ; 0.9163

26. log2 18 ; 4.1699 27. log9 6 ; 0.8155 28. log7 �8� ;

29. HORTICULTURE Siberian irises flourish when the concentration of hydrogen ions [H�]in the soil is not less than 1.58 � 10�8 mole per liter. What is the pH of the soil in whichthese irises will flourish? 7.8 or less

30. ACIDITY The pH of vinegar is 2.9 and the pH of milk is 6.6. How many times greater isthe hydrogen ion concentration of vinegar than of milk? about 5000

31. BIOLOGY There are initially 1000 bacteria in a culture. The number of bacteria doubleseach hour. The number of bacteria N present after t hours is N � 1000(2) t. How long willit take the culture to increase to 50,000 bacteria? about 5.6 h

32. SOUND An equation for loudness L in decibels is given by L � 10 log R, where R is thesound’s relative intensity. An air-raid siren can reach 150 decibels and jet engine noisecan reach 120 decibels. How many times greater is the relative intensity of the air-raidsiren than that of the jet engine noise? 1000

log10 8�2 log10 7

log10 6��log10 9

log10 18��log10 2

log10 9��log10 11

log10 32��log10 8

log10 12��log10 5

Practice (Average)

Common Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-410-4

0.5343

Reading to Learn MathematicsCommon Logarithms

NAME ______________________________________________ DATE ____________ PERIOD _____

10-410-4


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Pre-Activity Why is a logarithmic scale used to measure acidity?


Which substance is more acidic, milk or tomatoes? tomatoes

Reading the Lesson

1. Rhonda used the following keystrokes to enter an expression on her graphing calculator:

17

The calculator returned the result 1.230448921.Which of the following conclusions are correct? a, c, and d

a. The base 10 logarithm of 17 is about 1.2304.

b. The base 17 logarithm of 10 is about 1.2304.

c. The common logarithm of 17 is about 1.230449.

d. 101.230448921 is very close to 17.

e. The common logarithm of 17 is exactly 1.230448921.

2. Match each expression from the first column with an expression from the second columnthat has the same value.

a. log2 2 iv i. log4 1

b. log 12 iii ii. log2 8

c. log3 1 i iii. log10 12

d. log5 v iv. log5 5

e. log 1000 ii v. log 0.1

3. Calculators do not have keys for finding base 8 logarithms directly. However, you can use

a calculator to find log8 20 if you apply the formula.

Which of the following expressions are equal to log8 20? B and C

A. log20 8 B. C. D.


4. Sometimes it is easier to remember a formula if you can state it in words. State thechange of base formula in words. Sample answer: To change the logarithm of anumber from one base to another, divide the log of the original numberin the old base by the log of the new base in the old base.

log 8�log 20

log 20�log 8

log10 20�log10 8

change of base

1�5

ENTER) LOG


The Slide RuleBefore the invention of electronic calculators, computations were oftenperformed on a slide rule. A slide rule is based on the idea of logarithms. It hastwo movable rods labeled with C and D scales. Each of the scales is logarithmic.

To multiply 2 � 3 on a slide rule, move the C rod to the right as shownbelow. You can find 2 � 3 by adding log 2 to log 3, and the slide rule adds thelengths for you. The distance you get is 0.778, or the logarithm of 6.

Follow the steps to make a slide rule.

1. Use graph paper that has small squares, such as 10 squares to the inch. Using the scales shown at the right, plot the curve y � log x for x � 1, 1.5,and the whole numbers from 2 through 10. Make an obvious heavy dot for each point plotted.

2. You will need two strips of cardboard. A 5-by-7 index card, cut in half the long way,will work fine. Turn the graph you made in Exercise 1 sideways and use it to marka logarithmic scale on each of the twostrips. The figure shows the mark for 2 being drawn.

3. Explain how to use a slide rule to divide 8 by 2.

0

0.1

0.2

0.3 y

12

1 1.5 2

y = log x

0.1

0.2

1 2

1

21

CD

2

4

3

6

4 5 6 7 8 9

83 5 7 9

log 6

log 3log 2

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

C

D

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

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Study Guide and InterventionBase e and Natural Logarithms

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5

Base e and Natural Logarithms The irrational number e � 2.71828… often occursas the base for exponential and logarithmic functions that describe real-world phenomena.

Natural Base e As n increases, �1 � �napproaches e � 2.71828….

ln x � loge x

The functions y � ex and y � ln x are inverse functions.

Inverse Property of Base e and Natural Logarithms eln x � x ln ex � x

Natural base expressions can be evaluated using the ex and ln keys on your calculator.

Evaluate ln 1685.Use a calculator.ln 1685 � 7.4295

Write a logarithmic equation equivalent to e2x � 7.e2x � 7 → loge 7 � 2x or 2x � ln 7

Evaluate ln e18.Use the Inverse Property of Base e and Natural Logarithms.ln e18 � 18


1. ln 732 2. ln 84,350 3. ln 0.735 4. ln 1006.5958 11.3427 �0.3079 4.6052

5. ln 0.0824 6. ln 2.388 7. ln 128,245 8. ln 0.00614�2.4962 0.8705 11.7617 �5.0929

Write an equivalent exponential or logarithmic equation.

9. e15 � x 10. e3x � 45 11. ln 20 � x 12. ln x � 8ln x � 15 3x � ln 45 ex � 20 x � e8

13. e�5x � 0.2 14. ln (4x) � 9.6 15. e8.2 � 10x 16. ln 0.0002 � x�5x � ln 0.2 4x � e9.6 ln 10x � 8.2 ex � 0.0002


17. ln e3 18. eln 42 19. eln 0.5 20. ln e16.2

3 42 0.5 16.2

1�n

Example 1Example 1

Example 2Example 2

Example 3Example 3

ExercisesExercises


Equations and Inequalities with e and ln All properties of logarithms fromearlier lessons can be used to solve equations and inequalities with natural logarithms.


a. 3e2x � 2 � 103e2x � 2 � 10 Original equation

3e2x � 8 Subtract 2 from each side.

e2x � Divide each side by 3.

ln e2x � ln Property of Equality for Logarithms

2x � ln Inverse Property of Exponents and Logarithms

x � ln Multiply each side by �12

�.

x � 0.4904 Use a calculator.

b. ln (4x � 1) � 2

ln (4x � 1) � 2 Original inequality

eln (4x � 1) � e2 Write each side using exponents and base e.

0 � 4x � 1 � e2 Inverse Property of Exponents and Logarithms

1 � 4x � e2 � 1 Addition Property of Inequalities

� x � (e2 � 1) Multiplication Property of Inequalities

0.25 � x � 2.0973 Use a calculator.


1. e4x � 120 2. ex � 25 3. ex � 2 � 4 � 211.1969 {x|x � 3.2189} 4.8332

4. ln 6x � 4 5. ln (x � 3) � 5 � �2 6. e�8x � 50x � 9.0997 17.0855 {x |x � �0.4890}

7. e4x � 1 � 3 � 12 8. ln (5x � 3) � 3.6 9. 2e3x � 5 � 20.9270 6.7196 no solution

10. 6 � 3ex � 1 � 21 11. ln (2x � 5) � 8 12. ln 5x � ln 3x 90.6094 1492.9790 {x |x � 23.2423}

1�4

1�4

8�3

1�2

8�3

8�3

8�3


Base e and Natural Logarithms

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10-510-5

ExampleExample

ExercisesExercises

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5


1. e3 20.0855 2. e�2 0.1353

3. ln 2 0.6931 4. ln 0.09 �2.4079


5. ex � 3 x � ln 3 6. e4 � 8x 4 � ln 8x

7. ln 15 � x ex � 15 8. ln x � 0.6931 x � e0.6931


9. eln 3 3 10. eln 2x 2x

11. ln e�2.5 �2.5 12. ln ey y


13. ex � 5 {x |x � 1.6094} 14. ex � 3.2 {x |x � 1.1632}

15. 2ex � 1 � 11 1.7918 16. 5ex � 3 � 18 1.0986

17. e3x � 30 1.1337 18. e�4x 10 {x |x � �0.5756}

19. e5x � 4 34 {x |x � 0.6802} 20. 1 � 2e2x � �19 1.1513

21. ln 3x � 2 2.4630 22. ln 8x � 3 2.5107

23. ln (x � 2) � 2 9.3891 24. ln (x � 3) � 1 �0.2817

25. ln (x � 3) � 4 51.5982 26. ln x � ln 2x � 2 1.9221

Skills PracticeBase e and Natural Logarithms



1. e1.5 4.4817 2. ln 8 2.0794 3. ln 3.2 1.1632 4. e�0.6 0.5488

5. e4.2 66.6863 6. ln 1 0 7. e�2.5 0.0821 8. ln 0.037 �3.2968


9. ln 50 � x 10. ln 36 � 2x 11. ln 6 � 1.7918 12. ln 9.3 � 2.2300

ex � 50 e2x � 36 e1.7918 � 6 e2.2300 � 9.3

13. ex � 8 14. e5 � 10x 15. e�x � 4 16. e2 � x � 1

x � ln 8 5 � ln 10x x � �ln 4 2 � ln (x � 1)


17. eln 12 12 18. eln 3x 3x 19. ln e�1 �1 20. ln e�2y �2y


21. ex � 9 22. e�x � 31 23. ex � 1.1 24. ex � 5.8

{x |x � 2.1972} �3.4340 0.0953 1.7579

25. 2ex � 3 � 1 26. 5ex � 1 � 7 27. 4 � ex � 19 28. �3ex � 10 � 8

0.6931 {x |x � 0.1823} 2.7081 {x |x � �0.4055}

29. e3x � 8 30. e�4x � 5 31. e0.5x � 6 32. 2e5x � 24

0.6931 �0.4024 3.5835 0.4970

33. e2x � 1 � 55 34. e3x � 5 � 32 35. 9 � e2x � 10 36. e�3x � 7 � 15

1.9945 1.2036 0 {x |x � �0.6931}

37. ln 4x � 3 38. ln (�2x) � 7 39. ln 2.5x � 10 40. ln (x � 6) � 1

5.0214 �548.3166 8810.5863 8.7183

41. ln (x � 2) � 3 42. ln (x � 3) � 5 43. ln 3x � ln 2x � 9 44. ln 5x � ln x � 7

18.0855 145.4132 36.7493 14.8097

INVESTING For Exercises 45 and 46, use the formula for continuouslycompounded interest, A � Pert, where P is the principal, r is the annual interestrate, and t is the time in years.

45. If Sarita deposits $1000 in an account paying 3.4% annual interest compoundedcontinuously, what is the balance in the account after 5 years? $1185.30

46. How long will it take the balance in Sarita’s account to reach $2000? about 20.4 yr

47. RADIOACTIVE DECAY The amount of a radioactive substance y that remains after t years is given by the equation y � aekt, where a is the initial amount present and k isthe decay constant for the radioactive substance. If a � 100, y � 50, and k � �0.035,find t. about 19.8 yr

Practice (Average)

Base e and Natural Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

10-510-5

Reading to Learn MathematicsBase e and Natural Logarithms

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5

Pre-Activity How is the natural base e used in banking?


Suppose that you deposit $675 in a savings account that pays an annualinterest rate of 5%. In each case listed below, indicate which method ofcompounding would result in more money in your account at the end of oneyear.a. annual compounding or monthly compounding monthlyb. quarterly compounding or daily compounding dailyc. daily compounding or continuous compounding continuous

Reading the Lesson1. Jagdish entered the following keystrokes in his calculator:

5

The calculator returned the result 1.609437912. Which of the following conclusions arecorrect? d and fa. The common logarithm of 5 is about 1.6094.

b. The natural logarithm of 5 is exactly 1.609437912.

c. The base 5 logarithm of e is about 1.6094.

d. The natural logarithm of 5 is about 1.609438.

e. 101.609437912 is very close to 5.

f. e1.609437912 is very close to 5.

2. Match each expression from the first column with its value in the second column. Somechoices may be used more than once or not at all.

a. eln 5 IV I. 1

b. ln 1 V II. 10

c. eln e VI III. �1

d. ln e5 IV IV. 5

e. ln e I V. 0

f. ln � � III VI. e


3. A good way to remember something is to explain it to someone else. Suppose that you arestudying with a classmate who is puzzled when asked to evaluate ln e3. How would youexplain to him an easy way to figure this out? Sample answer: ln means naturallog. The natural log of e3 is the power to which you raise e to get e3. Thisis obviously 3.

1�e

ENTER) LN


Approximations for � and eThe following expression can be used to approximate e. If greater and greatervalues of n are used, the value of the expression approximates e more andmore closely.

�1 � �n1

��n

Another way to approximate e is to use this infinite sum. The greater thevalue of n, the closer the approximation.

e � 1 � 1 � �12� � �2

1� 3� � �2 �

13 � 4� � … � �2 � 3 � 4

1� … � n� � …

In a similar manner, � can be approximated using an infinite productdiscovered by the English mathematician John Wallis (1616–1703).

��2� � �

21� � �

23� � �

43� � �

45� � �

65� � �

67� � … � �2n

2�n

1� � �2n2�n

1� …

Solve each problem.

1. Use a calculator with an ex key to find e to 7 decimal places.

2. Use the expression �1 � �n1

��nto approximate e to 3 decimal places. Use

5, 100, 500, and 7000 as values of n.

3. Use the infinite sum to approximate e to 3 decimal places. Use the whole numbers from 3 through 6 as values of n.

4. Which approximation method approaches the value of e more quickly?

5. Use a calculator with a � key to find � to 7 decimal places.

6. Use the infinite product to approximate � to 3 decimal places. Use the whole numbers from 3 through 6 as values of n.

7. Does the infinite product give good approximations for � quickly?

8. Show that � 4 � � 5 is equal to e6 to 4 decimal places.

9. Which is larger, e� or � e?

10. The expression x reaches a maximum value at x � e. Use this fact to prove the inequality you found in Exercise 9.

Enrichment

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Study Guide and InterventionExponential Growth and Decay

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Less

on

10-

6Exponential Decay Depreciation of value and radioactive decay are examples ofexponential decay. When a quantity decreases by a fixed percent each time period, theamount of the quantity after t time periods is given by y � a(1 � r)t, where a is the initialamount and r is the percent decrease expressed as a decimal.Another exponential decay model often used by scientists is y � ae�kt, where k is a constant.

CONSUMER PRICES As technology advances, the price of manytechnological devices such as scientific calculators and camcorders goes down.One brand of hand-held organizer sells for $89.

a. If its price decreases by 6% per year, how much will it cost after 5 years?Use the exponential decay model with initial amount $89, percent decrease 0.06, andtime 5 years.y � a(1 � r)t Exponential decay formula

y � 89(1 � 0.06)5 a � 89, r � 0.06, t � 5

y � $65.32After 5 years the price will be $65.32.

b. After how many years will its price be $50?To find when the price will be $50, again use the exponential decay formula and solve for t.

y � a(1 � r)t Exponential decay formula

50 � 89(1 � 0.06)t y � 50, a � 89, r � 0.06

� (0.94)t Divide each side by 89.

log � � � log (0.94)t Property of Equality for Logarithms

log � � � t log 0.94 Power Property

t � Divide each side by log 0.94.

t � 9.3The price will be $50 after about 9.3 years.

1. BUSINESS A furniture store is closing out its business. Each week the owner lowersprices by 25%. After how many weeks will the sale price of a $500 item drop below $100?6 weeks

CARBON DATING Use the formula y � ae�0.00012t, where a is the initial amount ofCarbon-14, t is the number of years ago the animal lived, and y is the remainingamount after t years.

2. How old is a fossil remain that has lost 95% of its Carbon-14? about 25,000 years old

3. How old is a skeleton that has 95% of its Carbon-14 remaining? about 427.5 years old

log ��5809��

��log 0.94

50�89

50�89

50�89

ExampleExample

ExercisesExercises


Exponential Growth Population increase and growth of bacteria colonies are examplesof exponential growth. When a quantity increases by a fixed percent each time period, theamount of that quantity after t time periods is given by y � a(1 � r)t, where a is the initialamount and r is the percent increase (or rate of growth) expressed as a decimal.Another exponential growth model often used by scientists is y � aekt, where k is a constant.

A computer engineer is hired for a salary of $28,000. If she gets a5% raise each year, after how many years will she be making $50,000 or more?Use the exponential growth model with a � 28,000, y � 50,000, and r � 0.05 and solve for t.

y � a(1 � r)t Exponential growth formula

50,000 � 28,000(1 � 0.05)t y � 50,000, a � 28,000, r � 0.05

� (1.05)t Divide each side by 28,000.

log � � � log (1.05)t Property of Equality of Logarithms

log � � � t log 1.05 Power Property

t � Divide each side by log 1.05.

t � 11.9 years Use a calculator.

If raises are given annually, she will be making over $50,000 in 12 years.

1. BACTERIA GROWTH A certain strain of bacteria grows from 40 to 326 in 120 minutes.Find k for the growth formula y � aekt, where t is in minutes. about 0.0175

2. INVESTMENT Carl plans to invest $500 at 8.25% interest, compounded continuously.How long will it take for his money to triple? about 14 years

3. SCHOOL POPULATION There are currently 850 students at the high school, whichrepresents full capacity. The town plans an addition to house 400 more students. If the school population grows at 7.8% per year, in how many years will the new additionbe full? about 5 years

4. EXERCISE Hugo begins a walking program by walking mile per day for one week.

Each week thereafter he increases his mileage by 10%. After how many weeks is hewalking more than 5 miles per day? 24 weeks

5. VOCABULARY GROWTH When Emily was 18 months old, she had a 10-wordvocabulary. By the time she was 5 years old (60 months), her vocabulary was 2500 words.If her vocabulary increased at a constant percent per month, what was that increase?about 14%

1�2

log ��5208��

�log 1.05

50�28

50�28

50�28


Exponential Growth and Decay

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10-610-6

ExampleExample

ExercisesExercises

Skills PracticeExponential Growth and Decay

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6Solve each problem.

1. FISHING In an over-fished area, the catch of a certain fish is decreasing at an averagerate of 8% per year. If this decline persists, how long will it take for the catch to reachhalf of the amount before the decline? about 8.3 yr

2. INVESTING Alex invests $2000 in an account that has a 6% annual rate of growth. Tothe nearest year, when will the investment be worth $3600? 10 yr

3. POPULATION A current census shows that the population of a city is 3.5 million. Usingthe formula P � aert, find the expected population of the city in 30 years if the growthrate r of the population is 1.5% per year, a represents the current population in millions,and t represents the time in years. about 5.5 million

4. POPULATION The population P in thousands of a city can be modeled by the equationP � 80e0.015t, where t is the time in years. In how many years will the population of thecity be 120,000? about 27 yr

5. BACTERIA How many days will it take a culture of bacteria to increase from 2000 to50,000 if the growth rate per day is 93.2%? about 4.9 days

6. NUCLEAR POWER The element plutonium-239 is highly radioactive. Nuclear reactorscan produce and also use this element. The heat that plutonium-239 emits has helped topower equipment on the moon. If the half-life of plutonium-239 is 24,360 years, what isthe value of k for this element? about 0.00002845

7. DEPRECIATION A Global Positioning Satellite (GPS) system uses satellite informationto locate ground position. Abu’s surveying firm bought a GPS system for $12,500. TheGPS depreciated by a fixed rate of 6% and is now worth $8600. How long ago did Abubuy the GPS system? about 6.0 yr

8. BIOLOGY In a laboratory, an organism grows from 100 to 250 in 8 hours. What is thehourly growth rate in the growth formula y � a(1 � r) t? about 12.13%


Solve each problem.

1. INVESTING The formula A � P�1 � �2tgives the value of an investment after t years in

an account that earns an annual interest rate r compounded twice a year. Suppose $500is invested at 6% annual interest compounded twice a year. In how many years will theinvestment be worth $1000? about 11.7 yr

2. BACTERIA How many hours will it take a culture of bacteria to increase from 20 to2000 if the growth rate per hour is 85%? about 7.5 h

3. RADIOACTIVE DECAY A radioactive substance has a half-life of 32 years. Find theconstant k in the decay formula for the substance. about 0.02166

4. DEPRECIATION A piece of machinery valued at $250,000 depreciates at a fixed rate of12% per year. After how many years will the value have depreciated to $100,000?about 7.2 yr

5. INFLATION For Dave to buy a new car comparably equipped to the one he bought 8 yearsago would cost $12,500. Since Dave bought the car, the inflation rate for cars like his hasbeen at an average annual rate of 5.1%. If Dave originally paid $8400 for the car, howlong ago did he buy it? about 8 yr

6. RADIOACTIVE DECAY Cobalt, an element used to make alloys, has several isotopes.One of these, cobalt-60, is radioactive and has a half-life of 5.7 years. Cobalt-60 is used totrace the path of nonradioactive substances in a system. What is the value of k forCobalt-60? about 0.1216

7. WHALES Modern whales appeared 5�10 million years ago. The vertebrae of a whalediscovered by paleontologists contain roughly 0.25% as much carbon-14 as they wouldhave contained when the whale was alive. How long ago did the whale die? Use k � 0.00012. about 50,000 yr

8. POPULATION The population of rabbits in an area is modeled by the growth equationP(t) � 8e0.26t, where P is in thousands and t is in years. How long will it take for thepopulation to reach 25,000? about 4.4 yr

9. DEPRECIATION A computer system depreciates at an average rate of 4% per month. Ifthe value of the computer system was originally $12,000, in how many months is itworth $7350? about 12 mo

10. BIOLOGY In a laboratory, a culture increases from 30 to 195 organisms in 5 hours.What is the hourly growth rate in the growth formula y � a(1 � r) t? about 45.4%

r�2

Practice (Average)

Exponential Growth and Decay

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10-610-6

Reading to Learn MathematicsExponential Growth and Decay

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6Pre-Activity How can you determine the current value of your car?


• Between which two years shown in the table did the car depreciate bythe greatest amount?between years 0 and 1

• Describe two ways to calculate the value of the car 6 years after it waspurchased. (Do not actually calculate the value.)Sample answer: 1. Multiply $9200.66 by 0.16 and subtract theresult from $9200.66. 2. Multiply $9200.66 by 0.84.

Reading the Lesson

1. State whether each situation is an example of exponential growth or decay.

a. A city had 42,000 residents in 1980 and 128,000 residents in 2000. growth

b. Raul compared the value of his car when he bought it new to the value when hetraded ‘;lpit in six years later. decay

c. A paleontologist compared the amount of carbon-14 in the skeleton of an animalwhen it died to the amount 300 years later. decay

d. Maria deposited $750 in a savings account paying 4.5% annual interest compoundedquarterly. She did not make any withdrawals or further deposits. She compared thebalance in her passbook immediately after she opened the account to the balance 3 years later. growth

2. State whether each equation represents exponential growth or decay.

a. y � 5e0.15t growth b. y � 1000(1 � 0.05) t decay

c. y � 0.3e�1200t decay d. y � 2(1 � 0.0001) t growth


3. Visualizing their graphs is often a good way to remember the difference betweenmathematical equations. How can your knowledge of the graphs of exponential equationsfrom Lesson 10-1 help you to remember that equations of the form y � a(1 � r) t

represent exponential growth, while equations of the form y � a(1 � r) t representexponential decay?Sample answer: If a � 0, the graph of y � abx is always increasing if b � 1 and is always decreasing if 0 � b � 1. Since r is always a positivenumber, if b � 1 � r, the base will be greater than 1 and the function willbe increasing (growth), while if b � 1 � r, the base will be less than 1and the function will be decreasing (decay).


Effective Annual YieldWhen interest is compounded more than once per year, the effective annualyield is higher than the annual interest rate. The effective annual yield, E, isthe interest rate that would give the same amount of interest if the interestwere compounded once per year. If P dollars are invested for one year, thevalue of the investment at the end of the year is A � P(1 � E). If P dollarsare invested for one year at a nominal rate r compounded n times per year,

the value of the investment at the end of the year is A � P�1 � �nr

��n. Setting

the amounts equal and solving for E will produce a formula for the effectiveannual yield.

P(1 � E) � P�1 � �nr

��n

1 � E � �1 � �nr

��n

E � �1 � �nr

��n� 1

If compounding is continuous, the value of the investment at the end of oneyear is A � Per. Again set the amounts equal and solve for E. A formula forthe effective annual yield under continuous compounding is obtained.

P(1 � E) � Per

1 � E � er

E � er � 1

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Find the effectiveannual yield of an investment made at7.5% compounded monthly.r � 0.075

n � 12

E � �1 � �0.

10275��12

� 1 � 7.76%

Find the effectiveannual yield of an investment made at6.25% compounded continuously.r � 0.0625

E � e0.0625 � 1 � 6.45%


Find the effective annual yield for each investment.

1. 10% compounded quarterly 2. 8.5% compounded monthly

3. 9.25% compounded continuously 4. 7.75% compounded continuously

5. 6.5% compounded daily (assume a 365-day year)

6. Which investment yields more interest—9% compounded continuously or 9.2% compounded quarterly?

Chapter 10 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

Write the letter for the correct answer in the blank at the right of each question.

1. Find the domain and range of the function whose graph is shown.A. D � {x � x � 0}; R � {y � y � 0}B. D � {x � x is any real number.}; R � { y � y � 0}C. D � {x � x � 0}; R � {y � y is any real number.}D. D � {x � x is any real number.}; R � {y � y � 0} 1.

2. Which function represents exponential growth?

A. y � 9��13��

xB. y � 4x4 C. y � 12��

15��

xD. y � 10(2)x 2.

3. The graph of which exponential function passes through the points (0, 4) and (1, 24)?A. y � 4(6)x B. y � 3(8)x C. y � 2(2)x D. y � 10(3)x 3.

4. Simplify (x�7�)�3�.

A. x�21� B. x�7� � �3� C. x�10� D. x��73�� 4.

5. Solve 23m � 4 � 4.

A. m � 0 B. m � 0 C. m � 2 D. m � �53� 5.

6. Write the equation 43 � 64 in logarithmic form.A. log4 3 � 64 B. log3 4 � 64 C. log64 4 � 3 D. log4 64 � 3 6.

7. Write the equation log12 144 � 2 in exponential form.A. 1442 � 12 B. 122 � 144 C. 212 � 144 D. 14412 � 2 7.

8. Evaluate log2 8.A. 3 B. 4 C. 16 D. 64 8.

9. Solve log3 n � 2.A. 6 B. 5 C. 8 D. 9 9.

10. Solve log2 2m � log2 (m � 5).

A. m � �53� B. m � 5 C. m � 5 D. m � �5 10.

1010

y

xO

y � 4(2)x


Chapter 10 Test, Form 1 (continued)

11. Use log5 2 � 0.4307 to approximate the value of log5 4.A. 0.8614 B. 0.8980 C. 1.3652 D. 0.1855 11.

12. Solve log6 10 � log6 x � log6 40.A. 180 B. 4 C. 5 D. 30 12.

13. Solve 4x � 20. Round to four decimal places.A. 0.4628 B. 1.5214 C. 0.6990 D. 2.1610 13.

14. Solve 3x � 21. Round to four decimal places.A. x � 0.8451 B. x � 2.7712 C. x � 0.3608 D. x � 7.0000 14.

15. Express log9 22 in terms of common logarithms.

A. log �292� B. log 198 C. �

lloogg

292

� D. �lloogg

292� 15.

16. Evaluate eln 4.A. e4 B. 4e C. ln 4 D. 4 16.

17. Solve ex � 2.7.A. x � 0.9933 B. x � 0.9933 C. x � 1.0668 D. x � 1.0668 17.

18. Solve ln 3x � 1.A. 20.0855 B. 0.3333 C. 0.9061 D. 8.1548 18.

19. AUTOMOBILES Lydia bought a car for $20,000. It is expected to depreciate at a rate of 10% per year. What will be the value of the car in 2 years? Use y � a(1 � r)t and round to the nearest dollar.A. $16,200 B. $16,000 C. $19,980 D. $18,050 19.

20. ART Martin bought a painting for $5,000. It is expected to appreciate at 4% per year. How much will the painting be worth in 6 years? Use y � a(1 � r)t and round to the nearest cent.A. $6200.00 B. $5360.38 C. $37,647.68 D. $6326.60 20.

Bonus Evaluate 3 log2 64 � eln 5 � log�13�

9. B:

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Chapter 10 Test, Form 2A

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1. Find the domain and range of the function y � 3��15��

x.

A. D � {x � x is any real number.} B. D � {x � x is any real number.}R � { y � y � 0} R � { y � y � 0}

C. D � {x � x � 0} D. D � {x � x � 0}R � { y � y � 0} R � { y � y is any real number.} 1.

2. Which function represents exponential decay?

A. y � �1100�

(6)x B. y � (4x)�12�

C. y � 2��43��

xD. y � 12��

18��

x2.

3. Use the equation of the exponential function whose graph passes through the points (0, �3) and (2, �48) to find the value of y when x � �2.

A. ��34� B. ��

38� C. ��1

36�

D. 48 3.

4. Simplify m9�5� � m�5�.

A. m45 B. m9 C. m8�5� D. m10�5� 4.

5. Solve ��316��

n� 216n � 5.

A. 10 B. 3 C. �3 D. �10 5.

6. Solve 81y � 27y � 3

A. y � �9 B. y � 9 C. y � �9 D. y � 9 6.

7. Write the equation 6561�14�

� 9 in logarithmic form.

A. log�14�

9 � 6561 B. log6561 9 � �14�

C. log9 6561 � �14� D. log

�14�

6561 � 9 7.

8. Evaluate 5log5 63.A. 58 B. 315 C. log5 63 D. 63 8.

9. Solve log�15

�x � �1.

A. �215�

B. �5 C. 5 D. ��15� 9.

10. Solve log3 (5x � 1) � log3 (3x � 7)A. x � 3 B. x � 4 C. x 6 D. x � 27 10.

1010


Chapter 10 Test, Form 2A (continued)

11. Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the value of log5 54.A. 0.1370 B. 2.4785 C. 0.8820 D. 0.7488 11.

12. Solve log4 (m � 3) � log4 (m � 3) � 2.A. �11� B. 5 C. 1 D. �5.5 12.

13. Solve 63n � 435n � 4. Round to four decimal places.A. 1.1202 B. �1.9005 C. �0.2800 D. 2.1418 13.

14. Solve 52x � 1 � 50. Round to four decimal places.A. x � 4.5000 B. x � 0.7153 C. x � 0 D. x � 2.4307 14.

15. Use common logarithms to approximate log9 207 to four decimal places.A. 0.4120 B. 1.3617 C. 3.2702 D. 2.4270 15.

16. Evaluate ln e�9x.A. �9 ln x B. 9 ln x C. �9x D. 9x 16.

17. Solve 4 � 3e5x � 27.A. 0.4074 B. 0.4394 C. 2.0369 D. 0.1769 17.

18. Solve ln (x � 5) � 2.A. x � 2.3891 B. x 2.3891 C. x � 12.3891 D. x 12.3891 18.

19. CHEMISTRY A particular compound decays according to the equation y � ae�0.0974t, where t is in days. Find the half-life of this compound.A. about 5.1 days B. about 7.4 daysC. about 7.1 days D. about 9.7 days 19.

20. TOURISM At a town with a large convention center, the cost of a hotel room has increased 5.1% annually. If the average hotel room cost $48.00 in 1980 and this growth continues, what will an average hotel room cost in 2012? Use y � a(1 � r)t and round to the nearest cent.A. $143.38 B. $235.79 C. $126.34 D. $87.19 20.

Bonus Solve 5log5 2x � log5 (x � 3) � ln ex � 4. B:

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Chapter 10 Test, Form 2B

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1. Find the domain and range of the function y � �12�(2)x.

A. D � {x � x is any real number.} B. D � {x � x is any real number.}R � { y � y � 0} R � { y � y � 0}

C. D � {x � x � 0} D. D � {x � x � 0}R � { y � y � 0} R � { y � y is any real number.} 1.

2. Which function represents exponential growth?

A. y � �210��

52��

xB. y � 16(0.4)x C. y � �

12��

18��

xD. y � 8x3 2.

3. Use the equation of the exponential function whose graph passes through the points (0, �2) and (2, �50) to find the value of y when x � �2.

A. ��1100�

B. 50 C. ��225�

D. ��510�

3.

4. Simplify s7�11� � s�11�.

A. s77 B. s6�11� C. s8�11� D. s7 4.

5. Solve ��811��

t� 243t � 2.

A. �92� B. �

190� C. �

29� D. �1

90�

5.

6. Solve 64x � 32x � 2.A. x � �10 B. x � �10 C. x � 10 D. x � 10 6.

7. Write the equation log243 81 � �45� in exponential form.

A. 81�45�

� 243 B. 243�45�

� 81 C. ��45��

81� 243 D. ��

45��

243� 81 7.

8. Evaluate 9log9 54.A. log9 54 B. 54 C. 6 D. 486 8.

9. Solve log�18�

x � �1.

A. 8 B. �8 C. 0 D. ��18� 9.

10. Solve log2 (7x � 3) � log2 (x � 12).

A. x �52� B. x ��

52� C. x � �

32� D. x � �

52� 10.

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Chapter 10 Test, Form 2B (continued)

11. Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the value of log5 12.A. 0.8681 B. 0.1266 C. 1.5440 D. 0.5880 11.

12. Solve log3 a � log3 (a � 8) � 2.A. 8 B. 5 C. 9 D. �1, 9 12.

13. Solve 92n � 404n � 7. Round to four decimal places.A. 2.4922 B. 0.4012 C. �0.3560 D. �4.7209 13.

14. Solve 35x � 1 30. Round to four decimal places.A. x 0.4000 B. x 0.8192 C. x 1.8000 D. x 3.0959 14.

15. Use common logarithms to approximate log7 448 to four decimal places.A. 0.3188 B. 1.8062 C. 3.1372 D. �1.8062 15.

16. Evaluate eln (�6x).A. �6x B. 6 ln x C. 6x D. �6 ln x 16.

17. Solve ln (x � 2) � 3.A. 22.0855 B. 18.0855 C. 20.0855 D. �0.9014 17.

18. Solve e�9x 6.A. x � �1.8122 B. x 1.7918C. x � �0.08646 D. x � �0.1991 18.

19. CHEMISTRY A particular compound decays according to the equation y � ae�0.0736t, where t is in days. Find the half-life of the compound.A. about 9.1 days B. about 9.4 daysC. about 6.8 days D. about 7.4 days 19.

20. FOOD PRICES At a wholesale food distribution center, the price of sugar has increased 6.3% annually since 1980. Suppose sugar cost $0.43 per pound in 1980 and this growth continues. What will a pound of sugar cost in 2017? Use y � a(1 � r)t and round to the nearest cent.A. $4.12 B. $1.21 C. $2.42 D. $3.30 20.

Bonus Solve 2log2 5x � log2 (x � 1) � ln ex. B:

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Chapter 10 Test, Form 2C


1. Sketch the graph of y � �12�(3)x. Then state the function’s 1.

domain and range.

2. Determine whether the function y � 0.8��23��

xrepresents 2.

exponential growth or decay.

3. Write an exponential function whose graph passes through 3.the points (0, �6) and (�2, �54).

4. Simplify z5�7� z�7�. 4.

5. Solve �16� � 6n � 4. 5.

6. Solve 32x � 16x � 2. 6.

7. Write the equation log81 �19� � ��

12� in exponential form. 7.

8. Evaluate log9 97. 8.


10. Solve log36 n � �32�. 10.

11. Solve log5 (8x) � log5 (3x � 10). 11.


12. log5 48 12.

13. log5 �53� 13.

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Chapter 10 Test, Form 2C (continued)

For Questions 14–19, solve each equation or inequality.If necessary, round to four decimal places.

14. log4 n � �14� log4 81 � �

12� log4 25 14.

15. log2 (2x � 6) � log2 x � 3 15.

16. log3 (x � 3) � log3 (x � 2) � log3 14 16.

17. 6n � 2 � 50 17.

18. 2y � 5 y � 2 18.

19. 43x � 1 � 28 19.

20. Express log12 4 in terms of common logarithms. Then 20.approximate its value to four decimal places.

21. Evaluate ln e30. 21.

22. Solve ln (x � 5) � 3. 22.

23. Solve e�4x 9. 23.

24. CHEMISTRY After 12 hours, half of a 16-gram sample of a 24.radio-active element remains. Find the constant k for this element for t hours, then write the equation for modeling its exponential decay.

25. SAVINGS A savings account deposit of $150 is to earn 6.5% 25.interest. After how many years will the investment be worth $450? Use y � a(1 � r)t and round to the nearest tenth.

Bonus Evaluate (log4 123)(log12 43). B:

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Chapter 10 Test, Form 2D


1. Sketch the graph of y � 6��12��

x. Then state the function’s 1.

domain and range.

2. Determine whether the function y � 0.3��85��

xrepresents 2.

exponential growth or decay.

3. Write an exponential function whose graph passes through 3.the points (0, �5) and (�3, �40).

4. Simplify m9�6� � m�6�. 4.

5. Solve ��14��

m � 7� 16. 5.

6. Solve 10x � 3 � 100x � 1. 6.

7. Write the equation 5�4 � �6125�

in logarithmic form. 7.



10. Solve log64 x � �23�. 10.

11. Solve log4 (2x � 5) log4 (3x � 2). 11.


12. log5 18 12.

13. log5 �52� 13.

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Chapter 10 Test, Form 2D (continued)

For Questions 14–19, solve each equation or inequality.If necessary, round to four decimal places.

14. log5 n � �13� log5 64 � �

12� log5 49 14.

15. log6 (5 � 2a) � log6 (3a) � 1 15.

16. log3 (x � 3) � log3 (x � 2) � log3 6 16.

17. 7n � 3 � 80 17.

18. 3n � 6n � 2 18.

19. 54x � 1 � 30 19.


21. Evaluate eln 22. 21.

22. Solve ln (x � 4) � 4. 22.

23. Solve e�3x 18. 23.

24. CHEMISTRY In 5 years, radioactivity reduces the mass of 24.a 100-gram sample of an element to 75 grams. Find the constant k for this element for t in years, then write the equation for modeling this exponential decay.

25. SAVINGS A savings account deposit of $300 is to earn 5.8% 25.interest. After how many years will the investment be worth $900? Use y � a(1 � r)t and round to the nearest tenth.

Bonus Evaluate (log5 204)(log20 54). B:

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Chapter 10 Test, Form 3


1. Sketch the graph of y � �1.5(4)x. Then state the function’s 1.domain and range.

2. Determine whether the function y � 0.4(3.8)�x represents 2.exponential growth or decay.

3. Write an exponential function whose graph passes through 3.(0, �0.3) and (2, �10.8)

4. Solve 24x 321 � x � 8x � 2. 4.

5. Solve ��811��

4m � 1� ��2

17��

5m. 5.

6. Evaluate 2log2 (8x � 1). 6.

7. Evaluate log3 243x. 7.

8. Solve logx [log2 (log3 9)] � 2. 8.

9. Solve log3 (a2 � 12) � log3 a. 9.

For Questions 10 and 11, use log5 2 � 0.4307 andlog5 3 � 0.6826 to approximate the value of each expression.

10. log5 �145� 10.

11. log5 1.2 11.

12. Solve log4 0.25 � 3 log4 x � 5 log4 2 � �13� log4 64. 12.

13. Solve log4 (4b � 14) � log4 (b2 � 3b � 17) � �12�. 13.

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Chapter 10 Test, Form 3 (continued)

14. Solve loga (3n) � 2 loga x � loga x for n. 14.

15. Solve 4.5x2 � 2 � 32.7. Round to four decimal places. 15.

16. Solve 3n � �5n � 2�. Round to four decimal places. 16.

17. Solve ��12��

2t� 53 � 4t. Round to four decimal places. 17.

18. Express log5 (2.1)3 in terms of common logarithms. Then 18.approximate its value to four decimal places.

19. Evaluate e4 ln x. 19.

20. Solve ln (x � 3) � ln x � ln 4. 20.

21. Solve ln (x2 � 10) � ln x � ln 7. 21.

POPULATIONS For Questions 22 and 23, use the following information.

The population of Suffolk County in Massachusetts decreased from 663,906 in 1990 to 641,695 in 1999.

22. Write an exponential decay equation of the form y � aekt for 22.Suffolk County, where t is the number of years after 1990.

23. Use your equation to predict the population of Suffolk 23.County in 2020.

24. HOME OWNERSHIP The Richardson family bought a house 24.12 years ago for $95,000. The house is now worth $167,000.Assuming a steady rate of growth, what was the yearly rate of appreciation?

25. SCHOOL ENROLLMENT At a certain school, the number of 25.children entering kindergarten increased by 6.7% annually for 5 years and then decreased by 4.2% annually in the next 5 years. If 110 children enrolled in kindergarten at the beginning of this period, how many were enrolled after 10 years?

Bonus Solve log x2 � (log x)2. B:

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Chapter 10 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.

1. For the equation y � abx, where a � 0, we know that if b � 1, the function represents exponential growth, while it represents exponential decay if 0 � b � 1.a. Choose a positive value for a and let b � 1. Complete the table for these values

of a and b. Is y � abx an exponential function? Explain your reasoning.

b. Choose a positive value for a and a negative value for b. Complete the table for these values. Is y � abx an exponential function? Explain your reasoning.

2. a. Solve the exponential equation 35x � 9x � 6 by rewriting the equation so that each side has the same base.

b. Solve the equation in part a using common logarithms.c. Which method do you prefer? Explain your reasoning.d. Write and solve an exponential equation that you would also prefer to

solve using the method you chose in part c.

3. a. How are the three equations below alike? How are they different?log3 x � 2 log x � 2 ln x � 2

b. Solve each equation in part a above. Then write and solve a fourth equation that shares the same similarities and differences as the three given equations.

4. Ruby solved the exponential inequality 22z � 12z � 1 and stated that the solution set was {z � z � �2.2619}. When she checked her solution however, Ruby found that z � 1, which is in her solution set, does not make the original inequality true. When shechecked z � �3, which is not in her solution set, the original inequality is true.a. Show how Ruby arrived at her solution using common logarithms.b. What do her checks of z � 1 and z � �3 indicate about Ruby’s solution?c. What change must be made to the solution and why must that change be made?

5. ECONOMICS The Jones Corporation found that its annual profit could bemodeled by the exponential equation y � 10(0.99)t, while the Davis Company’sannual profit is modeled by y � 8(1.01)t. For both equations, profit is given inmillions of dollars, and t is the number of years since 1990.a. Find each company’s annual profit for the years between 1990 and 2000

to the nearest dollar.b. In which company would you prefer to own stock? Explain your reasoning.c. Indicate how a comparison of the two profit equations would support

your decision.

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x �3 �2 �1 0 1 2 3

y � abx

x �3 �2 �1 0 1 2 3

y � abx


Chapter 10 Vocabulary Test/Review

Choose from the terms above to complete each sentence.

1. A logarithm with base e is called a(n) .

2. The function y � 10x is an example of a(n) .

3. The equation y � e�0.2t is a model for .

4. The inverse of the function y � ex is the

5. The equation y � 100(1 � 0.1)t is a model for .

6. The value of log3 50 can be found by using the with a calculator.

7. y � log2 x is an example of a .

8. 5x � 1 � 125 and 9x � 272x � 1 are examples of .

9. A logarithm with base 10 is called a(n) .

10. In the equation y � 20(1 � 0.02)t, 0.02 is called the .

In your own words—Define each term.

11. logarithm

12. natural base, e

Change of Base Formulacommon logarithmexponential decayexponential equationexponential function

exponential growthexponential inequalitylogarithmlogarithmic function

natural base, enatural base exponential function

natural logarithm

natural logarithmic function

rate of decayrate of growth

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Chapter 10 Quiz (Lessons 10–1 and 10–2)

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1. Sketch the graph of y � 3��12��

x. Then state the function’s

domain and range.

2. Write an exponential function whose graph passes through the points (0, �5) and (�2, �20). Then determine whether the function represents exponential growth or decay.

3. Simplify 3�5� 32�5�. 3.

4. Solve ��13��

m� 27m � 2. 4.

5. Solve 254t � 1 � 1252t. 5.

6. Write the equation 81�12�

� 9 in logarithmic form. 6.

7. Write the equation log216 36 � �23� in exponential form. 7.


9. Solve log16 n � ��12�. 9.

10. Solve log5 (4x � 1) � log5 (x � 2). 10.

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Chapter 10 Quiz (Lesson 10–3)

Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the value of each expression. 1.

1. log5 �130� 2. log5 24 2.

Solve each equation.

3. log7 36 � log7 (2x) � log7 4 3.

4. log3 x � �12� log3 25 � 5 log3 2 4.

5. log2 (x � 1) � log2 (x � 5) � 4 5.

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1. y

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1. log 1.5 2. ln 4.1

For Questions 3–7, solve each equation or inequality.Round to four decimal places.

3. 42m � 130 4. 5x � 4 � 23x

5. 7t � 5 � 21.5 6. ln (x � 5) � 3

7. 4 � 2e5x � 28 7.

8. Express log3 25 in terms of common logarithms. Then approximate its value to four decimal places. 8.

9. Write an equivalent logarithmic equation for e3 � 2x. 9.

10. Evaluate eln 0.3. 10.

Chapter 10 Quiz (Lesson 10–6)

1. A substance decays according to the equation y � ae�0.0025t, 1.where t is in minutes. Find the half-life of the substance.Round to the nearest tenth.

2. A-1 Electric has a piece of machinery valued at $55,000. 2.It depreciates at a rate of 12.5% per year. After how many years will the value have depreciated to $38,000? Round to the nearest tenth.

3. Standardized Test Practice In 1925, the population of acity was 90,000. Since then, the population has increased by 2.1% per year. If it continues to grow at this rate, what will the population be in 2020? A. 4,073,333 B. 136,382 C. 648,169 D. 6.6 � 1012 3.

4. The Morgans bought a house worth $125,000. Assuming 4.that the house will appreciate 8% per year, what will the house be worth in eight years? Round to the nearest dollar.

5. A type of bacteria doubles in number every 25 minutes. 5.Find the constant k for this type of bacteria, then write the equation for modeling this exponential growth.

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Chapter 10 Quiz (Lessons 10–4 and 10–5)

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1.

2.

3.

4.

5.

6.

Chapter 10 Mid-Chapter Test (Lessons 10–1 through 10–4)



1. Find the domain and range of the function shown.A. D � {x � x � 0}, R � { y � y is any real number.}B. D � {x � x is any real number.}, R � { y � y � 0}C. D � {x � x is any real number.}, R � { y � y � 0}D. D � {x � x � 0}, R � { y � y � 0} 1.

2. Simplify the expression y5�7� � y�7�.

A. y35 B. y5 C. y6�7� D. y4�7� 2.

3. Write the equation 4�3 � �614�

in logarithmic form.

A. log�3 4 � �614�

B. log4 �614�

� �3

C. log�614�

(�3) � 4 D. log4 (�3) � �614�

3.

4. Evaluate log4 32.

A. �52� B. 8 C. 3 D. �

25� 4.

5. Solve log3 (7x � 3) � log3 (5x).

A. x � �32� B. x � �

37� C. x � 0 D. x � �

23� 5.

6. Use log3 5 � 1.4650 and log3 7 � 1.7712 to approximate the value of log3 �251�.

A. 3.6270 B. 3.8486 C. 1.8916 D. 1.3062 6.

7. Write an exponential function whose graph passes through 7.the points (0, �3) and (4, �48).

For Questions 9–13, solve each equation or inequality. 8.

8. ��18��

x� 4x � 5 9. log

�15�

m � �2 9.

10. log7 (x � 3) � log7 (x � 3) � 1 10.

11. log3 (y � 8) � log3 (y � 4) � log3 13 11.

12. Use log2 3 � 1.5850 and log2 7 � 2.8074 to approximate 12.the value of log2 84.

Part II

Part I

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Chapter 10 Cumulative Review (Chapters 1–10)

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1. Name the sets of numbers to which ��16� belongs. (Lesson 1–2)

For Questions 2 and 3, use the following information.A new camp site will contain t tent sites, with 25 square meters of land each, and r recreational vehicle (RV) sites with 40 square meters of land each. No more than 90 camp sites can be built on the 3000 square meters of land available.

2. Write a system of inequalities to represent the number of sites built. Then list the coordinates of the vertices of the feasible region. (Lesson 3–4)

3. The site owner will charge $14 per tent site and $20 per RV site per day. Write a function for the total profit per day. Then determine the number of each type of site needed to earn a maximum profit, and find the maximum profit per day. (Lesson 3–4)

4. Simplify �3

27x3y�. (Lesson 5–6)

5. Solve 2x2 � x � 1 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. (Lesson 6–2)

6. Find all the zeros of the function h(x) � x3 � 5x2 � 8x � 6.(Lesson 7–5)

7. Find (f � g)(x), (f � g)(x), (f g)(x), and ��gf��(x) for

f(x) � x2 � 2x � 15 and g(x) � 2x � 1. (Lesson 7–7)

8. Write an equation for the hyperbola with vertices (0, 5) and (0, �5) and a conjugate axis at length 6 units. (Lesson 8–5)

9. Find the exact solution(s) of the system of equations x2 � y2 � 13x2 � 8y2 � 4. (Lesson 8–7)

For Questions 10 and 11, simplify.

10. �10

sm2n4

� � ��5ms3

n��

2(Lesson 9–1)

11. �x �

45

� � �5 �

3x

� (Lesson 9–2)

12. Identify the function represented by the equation y � �2x . (Lesson 9–5)

13. Write an exponential function whose graph passes through the points (0, 4) and (�2, 100). (Lesson 10–1)

14. Solve log�19

�x � �2. (Lesson 10–2)

15. Solve 65n � 542n � 3. Round to four decimal places. (Lesson 10–4)

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

y

xO

Ass

essm

ent

Standardized Test Practice (Chapters 1–10)


NAME DATE PERIOD

1010

1. If �8x

� � x, which could be a value for x?

A. �1 B. 0 C. 2 D. �14

� 1.

2. If 0 � a � 1, which of the following increases as a decreases?

E. a � 1 F. a2 � 1 G. �a1

� H. a2 2.

3. If 3x � 2 is an odd integer, what is the next consecutive odd integer?A. 3x � 1 B. 3x � 3 C. 3x � 1 D. 3x 3.

4. Jody sold 4 more than twice the number of cars that Laura sold.If Laura sold c cars, how many more did Jody sell than Laura?E. 4 F. c � 4 G. 3c � 4 H. 2c � 4 4.

5. If 8 � 3z � 16 � 5z, then what is the value of 4z?A. �16 B. �4 C. 1 D. 12 5.

6. The radius of a wheel is 6 inches. How many revolutions will it make if it is rolled a distance of 288� inches?E. 8 F. 8� G. 24 H. 24� 6.

7. What is the 8th term in the sequence 3, 2, 0, �4, �12, …?A. �124 B. �60 C. �36 D. �144 7.

8. Which Venn diagram models the relationships among the sets A � {1, 2, 3}, B � {�4, 0}, and C � {positive integers}?E. F. G. H. 8.

9. A total of $270 is to be divided among four children. Each will receive an amount that is proportional to his or her age. If the children are 5, 10, 14, and 16 years old, how much money does the youngest child receive?A. $96 B. $6 C. $30 D. $54 9. DCBA

HGFE

B

A

CBA

C

BA

C

BA

C

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.


Standardized Test Practice (continued)

NAME DATE PERIOD

1010

NAME DATE PERIOD

10. If m2 � n2 � 140 and mn � 49, what is the value of (m � n)2?

11. What is the slope of a line that is perpendicular to the graph of 5x � 4y � 7?

12. If � � m in the figure shown,what is the value of d?

13. Find the perimeter of square EFGH if the areas of rectangle ABCD and square EFGH are equal.

Column A Column B

14. 14.

15. A right triangle has sides 6, 6, and c. 15.

16. �x32� � y 16.

17. 17.

18. S � {19, 22, 11, 17, 35} 18.

The mean of set SThe median of set S

DCBA

DCBA(�3)333(�3)104

yx

DCBA

c8

DCBA

DCBA14% of 230023% of 1400

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

10. 11.

12. 13.

0 0 0

.. ./ /

.

99 9 987654321

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.

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Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

d˚60˚

132˚m

�

A B

D

E F

H G

C

x � 18

2x

x

A

D

C

B

Unit 3 Test (Chapters 8–10)


1. Find the midpoint of the line segment with end points at 1.(�6, 2) and (3, 8).

2. Write an equation for the parabola with focus (4, 0) and 2.directrix y � 2.

3. Graph 16x2 � 9y2 � 144. 3.

4. Graph �x9

2� � �

(y �

92)2� � 1. 4.

5. Write an equation for the circle with center (�4, �7�) and 5.radius 5 units.

6. Write an equation for the ellipse with end points of the 6.major axis at (7, 1) and (�7, 1) and end points of the minor axis at (0, 5) and (0, �3).

7. Write 36x2 � 360x � 25y2 � 100y � 100 in standard form. 7.Then state whether the graph of the equation is a parabola,circle, ellipse, or hyperbola.

8. Find the exact solutions of the system of equations 8.x2 � 2y2 � 18 and x � 2y.

9. Simplify �4zx

2y3� � ��8z

x3

2y��2

. 9.

10. Simplify �d2d� 9� � �2d

5� 6�. 10.

11. Find the LCM of m2 � 4m � 5 and m2 � 8m � 7. 11.

12. Determine the equations of any vertical asymptotes and the 12.

value of x for any holes in the graph of f(x) ��x2 �

x1�

1x2� 18

�.

y

xO

y

xO

NAME DATE PERIOD

SCORE

Ass

essm

ent


Unit 3 Test (continued)(Chapters 8–10)

13. If y varies jointly as x and z and y � 100 when x � 10 and 13.z � 5, find y when x � 12 and z � 6.

14. Identify the type of function represented 14.by the graph.

15. Identify the function represented by y � �3x

�. 15.

16. Solve �t �

85

� � �tt

�

�

35

� � �13

�. 16.

17. Sketch the graph of y � 1.5(2)x. Then state the function’s 17.domain and range.

18. Determine whether y � 1.5��16��x

represents exponential 18.growth or decay.

19. Simplify x5 x�. 19.

For Questions 20–24, solve each equation or inequality.Round to four decimal places if necessary.

20. ��15

��t�2� 125 21. log4 (x � 9) � 2

22. log4 z � log4 (z � 3) � 1

23. 3.9m�4 � 10.21 24. e3x � 21


26. Evaluate ln e�3x. 26.

27. Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the 27.value of log5 12.


29. In a certain area, the sale price of new single-family homes 29.has increased 4.1% per year since 1992. If a house was purchased in this area in 1992 for $75,000 and the growth continues, what will the sale price be in 2006? Use y � a(1 � r)t and round to the nearest cent.

y

xO

NAME DATE PERIOD

20.

21.

22.

23.

24.

y

xO

Standardized Test PracticeStudent Record Sheet (Use with pages 572–573 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

1010

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

For Questions 12–18, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

11 13 15 17

12 14 16 18

Select the best answer from the choices given and fill in the corresponding oval.

19 21 23

20 22 DCBADCBA

DCBADCBADCBA

0 0 0

.. ./ /

.

99 9 987654321

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.

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99 9 987654321

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.

99 9 987654321

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.

99 9 987654321

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0 0 0

.. ./ /

.

99 9 987654321

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0 0 0

.. ./ /

.

99 9 987654321

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87654321

87654321

DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


Answers (Lesson 10-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

Exp

on

enti

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1

©G

lenc

oe/M

cGra

w-H

ill57

3G

lenc

oe A

lgeb

ra 2

Lesson 10-1

Exp

on

enti

al F

un

ctio

ns

An

exp

onen

tial

fu

nct

ion

has

th

e fo

rm y

�ab

x ,w

her

e a

�0,

b�

0,an

d b

�1.

1.T

he f

unct

ion

is c

ontin

uous

and

one

-to-

one.

Pro

per

ties

of

an2.

The

dom

ain

is t

he s

et o

f al

l rea

l num

bers

.

Exp

on

enti

al F

un

ctio

n3.

The

x-a

xis

is t

he a

sym

ptot

e of

the

gra

ph.

4.T

he r

ange

is t

he s

et o

f al

l pos

itive

num

bers

if a

�0

and

all n

egat

ive

num

bers

if a

�0.

5.T

he g

raph

con

tain

s th

e po

int

(0,

a).

Exp

on

enti

al G

row

thIf

a�

0 an

d b

�1,

the

fun

ctio

n y

�ab

xre

pres

ents

exp

onen

tial g

row

th.

and

Dec

ayIf

a�

0 an

d 0

�b

�1,

the

fun

ctio

n y

�ab

xre

pres

ents

exp

onen

tial d

ecay

.

Sk

etch

th

e gr

aph

of

y�

0.1(

4)x .

Th

en s

tate

th

e

fun

ctio

n’s

dom

ain

an

d r

ange

.M

ake

a ta

ble

of v

alu

es.C

onn

ect

the

poin

ts t

o fo

rm a

sm

ooth

cu

rve.

Th

e do

mai

n o

f th

e fu

nct

ion

is

all

real

nu

mbe

rs,w

hil

e th

e ra

nge

is

the

set

of a

ll p

osit

ive

real

nu

mbe

rs.

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

rep

rese

nts

exp

onen

tial

g

row

thor

dec

ay.

a.y

�0.

5(2)

xb

.y�

�2.

8(2)

xc.

y�

1.1(

0.5)

x

expo

nen

tial

gro

wth

,n

eith

er,s

ince

�2.

8,ex

pon

enti

al d

ecay

,sin

cesi

nce

th

e ba

se,2

,is

the

valu

e of

ais

les

s th

e ba

se,0

.5,i

s be

twee

ngr

eate

r th

an 1

than

0.

0 an

d 1

Sk

etch

th

e gr

aph

of

each

fu

nct

ion

.Th

en s

tate

th

e fu

nct

ion

’s d

omai

n a

nd

ran

ge.

1.y

�3(

2)x

2.y

��

2 ��x

3.y

�0.

25(5

)x

Do

mai

n:

all r

eal

Do

mai

n:

all r

eal

Do

mai

n:

all r

eal

nu

mb

ers;

Ran

ge:

all

nu

mb

ers;

Ran

ge:

all

nu

mb

ers;

Ran

ge:

all

po

siti

ve r

eal n

um

ber

sn

egat

ive

real

nu

mb

ers

po

sitiv

e re

al n

um

ber

sD

eter

min

e w

het

her

eac

h f

un

ctio

n r

epre

sen

ts e

xpon

enti

al g

row

th o

r d

ecay

.

4.y

�0.

3(1.

2)x

gro

wth

5.y

��

5 ��x

nei

ther

6.y

�3(

10)�

xd

ecay

4 � 5

x

y

O

x

y

O

x

y

O

1 � 4

x�

10

12

3

y0.

025

0.1

0.4

1.6

6.4

x

y

O

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill57

4G

lenc

oe A

lgeb

ra 2

Exp

on

enti

al E

qu

atio

ns

and

Ineq

ual

itie

sA

ll t

he

prop

erti

es o

f ra

tion

al e

xpon

ents

that

you

kn

ow a

lso

appl

y to

rea

l ex

pon

ents

.Rem

embe

r th

at a

m�

an�

am

n,(

am)n

�am

n,

and

am

an�

am�

n.

Pro

per

ty o

f E

qu

alit

y fo

rIf

bis

a p

ositi

ve n

umbe

r ot

her

than

1,

Exp

on

enti

al F

un

ctio

ns

then

bx

�b

yif

and

only

if x

�y.

Pro

per

ty o

f In

equ

alit

y fo

rIf

b�

1

Exp

on

enti

al F

un

ctio

ns

then

bx

�b

yif

and

only

if x

�y

and

bx

�b

yif

and

only

if x

�y.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Exp

on

enti

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1

Sol

ve 4

x�

1�

2x

�5 .

4x�

1�

2x

5O

rigin

al e

quat

ion

(22 )

x�

1�

2x

5R

ewrit

e 4

as 2

2 .

2(x

�1)

�x

5

Pro

p. o

f In

equa

lity

for

Exp

onen

tial

Fun

ctio

ns

2x�

2 �

x

5D

istr

ibut

ive

Pro

pert

y

x�

7S

ubtr

act

xan

d ad

d 2

to e

ach

side

.

Sol

ve 5

2x�

1�

.

52x

�1

�O

rigin

al in

equa

lity

52x

�1

�5�

3R

ewrit

e as

5�

3 .

2x�

1 �

�3

Pro

p. o

f In

equa

lity

for

Exp

onen

tial F

unct

ions

2x�

�2

Add

1 t

o ea

ch s

ide.

x�

�1

Div

ide

each

sid

e by

2.

Th

e so

luti

on s

et i

s {x

|x�

�1}

.

1� 12

5

1� 12

5

1� 12

5Ex

ampl

e1Ex

ampl

e1Ex

ampl

e2Ex

ampl

e2

Exer

cises

Exer

cises

Sim

pli

fy e

ach

exp

ress

ion

.

1.(3

�2� )�

2�2.

25�

2��

125�

2�3.

(x�2� y

3�2� )�

2�

955

�2�

or

3125

�2�

x2 y

6

4.(x�

6� )(x�

5� )5.

(x�6� )�

5�6.

(2x�

)(5x3

�)

x�6�

��

5�x�

30�10

x4�

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ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

on.

7.32

x�

1�

3x

23

8.23

x�

4x

24

9.32

x�

1�

�

10.4

x

1�

82x

3

�11

.8x

�2

�12

.252

x�

125x

2

6

13.4

�x�

�16

�5�

2014

.x�

3��

36��3 4�

615

.x�

2��

81� �1 8��

3

16.3

x�

4�

x�

117

.42x

�2

�2x

1

x�

18.5

2x�

125x

�5

x�

15

19.1

04x

1

�10

0x�

220

.73x

�49

x221

.82x

�5

�4x

8

x�

�x

�o

r x

�0

x�

�3 41 �3 � 2

5 � 2

5 � 31 � 27

2 � 31 � 16

7 � 4

1 � 21 � 9


A


Skil

ls P

ract

ice

Exp

on

enti

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1

©G

lenc

oe/M

cGra

w-H

ill57

5G

lenc

oe A

lgeb

ra 2

Lesson 10-1

Sk

etch

th

e gr

aph

of

each

fu

nct

ion

.Th

en s

tate

th

e fu

nct

ion

’s d

omai

n a

nd

ran

ge.

1.y

�3(

2)x

2.y

�2 �

�x

do

mai

n:

all r

eal n

um

ber

s;d

om

ain

:al

l rea

l nu

mb

ers;

ran

ge:

all p

osi

tive

nu

mb

ers

ran

ge:

all p

osi

tive

nu

mb

ers

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

rep

rese

nts

exp

onen

tial

gro

wth

or d

eca

y.

3.y

�3(

6)x

gro

wth

4.y

�2 �

�xd

ecay

5.y

�10

�x

dec

ay6.

y�

2(2.

5)x

gro

wth

Wri

te a

n e

xpon

enti

al f

un

ctio

n w

hos

e gr

aph

pas

ses

thro

ugh

th

e gi

ven

poi

nts

.

7.(0

,1)

and

(�1,

3)y

��

�x8.

(0,4

) an

d (1

,12)

y�

4(3)

x

9.(0

,3)

and

(�1,

6)y

�3 �

�x10

.(0,

5) a

nd

(1,1

5)y

�5(

3)x

11.(

0,0.

1) a

nd

(1,0

.5)

y�

0.1(

5)x

12.(

0,0.

2) a

nd

(1,1

.6)

y�

0.2(

8)x

Sim

pli

fy e

ach

exp

ress

ion

.

13. (3

�3� )�

3�27

14.(

x�2� )�

7�x�

14�

15.5

2�3�

�54

�3�

56�

3�16

.x3�

x�

x2�

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ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

on.

17.3

x�

9x

�2

18.2

2x

3�

321

19.4

9x�

x

�20

.43x

�2

�16

21.3

2x

5�

27x

522

.27x

�32

x

334 � 3

1 � 21 � 7

1 � 2

1 � 3

9 � 10

x

y

Ox

y

O

1 � 2

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lenc

oe/M

cGra

w-H

ill57

6G

lenc

oe A

lgeb

ra 2

Sk

etch

th

e gr

aph

of

each

fu

nct

ion

.Th

en s

tate

th

e fu

nct

ion

’s d

omai

n a

nd

ran

ge.

1.y

�1.

5(2)

x2.

y�

4(3)

x3.

y�

3(0.

5)x

do

mai

n:

all r

eal

do

mai

n:

all r

eal

do

mai

n:

all r

eal

nu

mb

ers;

ran

ge:

all

nu

mb

ers;

ran

ge:

all

nu

mb

ers;

ran

ge:

all

po

siti

ve n

um

ber

sp

osi

tive

nu

mb

ers

po

siti

ve n

um

ber

s

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

rep

rese

nts

exp

onen

tial

gro

wth

or d

eca

y.

4.y

�5(

0.6)

xd

ecay

5.y

�0.

1(2)

xg

row

th6.

y�

5 �

4�x

dec

ay

Wri

te a

n e

xpon

enti

al f

un

ctio

n w

hos

e gr

aph

pas

ses

thro

ugh

th

e gi

ven

poi

nts

.

7.(0

,1)

and

(�1,

4)8.

(0,2

) an

d (1

,10)

9.(0

,�3)

an

d (1

,�1.

5)

y�

��x

y�

2(5)

xy

��

3(0.

5)x

10.(

0,0.

8) a

nd

(1,1

.6)

11.(

0,�

0.4)

an

d (2

,�10

)12

.(0,

�)

and

(3,8

�)

y�

0.8(

2)x

y�

�0.

4(5)

xy

��

(2)x

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pli

fy e

ach

exp

ress

ion

.

13.(2

�2� )�

8�16

14.(

n�

3� )�75�

n15

15.y

�6�

�y5

�6�

y6�

6�

16.1

3�6�

�13

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133�

6�17

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n

�n

3 �

�18

.125

�11�

5�

11�52

�11�

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

on.

19.3

3x �

5�

81x

�3

20.7

6x�

72x

�20

�5

21.3

6n�

5�

94n

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n�

22.9

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1�

27x

4

1423

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�nn

24

.164

n�

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1282

n

1

BIO

LOG

YF

or E

xerc

ises

25

and

26,

use

th

e fo

llow

ing

info

rmat

ion

.T

he

init

ial

nu

mbe

r of

bac

teri

a in

a c

ult

ure

is

12,0

00.T

he

nu

mbe

r af

ter

3 da

ys i

s 96

,000

.

25.W

rite

an

exp

onen

tial

fu

nct

ion

to

mod

el t

he

popu

lati

on y

of b

acte

ria

afte

r x

days

.y

�12

,000

(2)x

26.H

ow m

any

bact

eria

are

th

ere

afte

r 6

days

?76

8,00

0

27.E

DU

CA

TIO

NA

col

lege

wit

h a

gra

duat

ing

clas

s of

400

0 st

ude

nts

in

th

e ye

ar 2

002

pred

icts

th

at i

t w

ill

hav

e a

grad

uat

ing

clas

s of

486

2 in

4 y

ears

.Wri

te a

n e

xpon

enti

alfu

nct

ion

to

mod

el t

he

nu

mbe

r of

stu

den

ts y

in t

he

grad

uat

ing

clas

s t

year

s af

ter

2002

.y

�40

00(1

.05)

t

11 � 21 � 6

1 � 8

1 � 2

1 � 4

x

y

Ox

y

Ox

y

OPra

ctic

e (

Ave

rag

e)

Exp

on

enti

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1



Readin

g t

o L

earn

Math

em

ati

csE

xpo

nen

tial

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1

©G

lenc

oe/M

cGra

w-H

ill57

7G

lenc

oe A

lgeb

ra 2

Lesson 10-1

Pre-

Act

ivit

yH

ow d

oes

an e

xpon

enti

al f

un

ctio

n d

escr

ibe

tou

rnam

ent

pla

y?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-1 a

t th

e to

p of

pag

e 52

3 in

you

r te

xtbo

ok.

How

man

y ro

un

ds o

f pl

ay w

ould

be

nee

ded

for

a to

urn

amen

t w

ith

100

play

ers?

7

Rea

din

g t

he

Less

on

1.In

dica

te w

het

her

eac

h o

f th

e fo

llow

ing

stat

emen

ts a

bou

t th

e ex

pon

enti

al f

un

ctio

n

y�

10x

is t

rue

or f

alse

.

a.T

he

dom

ain

is

the

set

of a

ll p

osit

ive

real

nu

mbe

rs.

fals

e

b.

Th

e y-

inte

rcep

t is

1.

tru

e

c.T

he

fun

ctio

n i

s on

e-to

-on

e.tr

ue

d.

Th

e y-

axis

is

an a

sym

ptot

e of

th

e gr

aph

.fa

lse

e.T

he

ran

ge i

s th

e se

t of

all

rea

l n

um

bers

.fa

lse

2.D

eter

min

e w

het

her

eac

h f

un

ctio

n r

epre

sen

ts e

xpon

enti

al g

row

thor

dec

ay.

a.y

�0.

2(3)

x .g

row

thb

.y�

3 ��x .

dec

ayc.

y�

0.4(

1.01

)x.

gro

wth

3.S

upp

ly t

he

reas

on f

or e

ach

ste

p in

th

e fo

llow

ing

solu

tion

of

an e

xpon

enti

al e

quat

ion

.

92x

�1

�27

xO

rigi

nal

equ

atio

n

(32 )

2x�

1�

(33 )

xR

ewri

te e

ach

sid

e w

ith

a b

ase

of

3.32

(2x

�1)

�33

xP

ow

er o

f a

Po

wer

2(2x

�1)

�3x

Pro

per

ty o

f E

qu

alit

y fo

r E

xpo

nen

tial

Fu

nct

ion

s4x

�2

�3x

Dis

trib

uti

ve P

rop

erty

x�

2 �

0S

ub

trac

t 3x

fro

m e

ach

sid

e.x

�2

Ad

d 2

to

eac

h s

ide.

Hel

pin

g Y

ou

Rem

emb

er

4.O

ne

way

to

rem

embe

r th

at p

olyn

omia

l fu

nct

ion

s an

d ex

pon

enti

al f

un

ctio

ns

are

diff

eren

tis

to

con

tras

t th

e po

lyn

omia

l fu

nct

ion

y�

x2an

d th

e ex

pon

enti

al f

un

ctio

n y

�2x

.Tel

l at

leas

t th

ree

way

s th

ey a

re d

iffe

ren

t.

Sam

ple

an

swer

:In

y�

x2 ,

the

vari

able

xis

a b

ase,

but

in y

�2x

,th

eva

riab

le x

is a

n e

xpo

nen

t.T

he

gra

ph

of

y�

x2

is s

ymm

etri

c w

ith

res

pec

tto

th

e y-

axis

,bu

t th

e g

rap

h o

f y

�2x

is n

ot.

Th

e g

rap

h o

f y

�x

2to

uch

esth

e x-

axis

at

(0,0

),bu

t th

e g

rap

h o

f y

�2x

has

th

e x-

axis

as

an a

sym

pto

te.

You

can

co

mp

ute

th

e va

lue

of

y�

x2

men

tally

fo

r x

�10

0,bu

t yo

u c

ann

ot

com

pu

te t

he

valu

e o

f y

�2x

men

tally

fo

r x

�10

0.

2 � 5

©G

lenc

oe/M

cGra

w-H

ill57

8G

lenc

oe A

lgeb

ra 2

Fin

din

g S

olu

tio

ns

of

xy�

yx

Per

hap

s yo

u h

ave

not

iced

th

at i

f x

and

yar

e in

terc

han

ged

in e

quat

ion

s su

chas

x�

yan

d xy

�1,

the

resu

ltin

g eq

uat

ion

is

equ

ival

ent

to t

he

orig

inal

equ

atio

n.T

he

sam

e is

tru

e of

th

e eq

uat

ion

xy

�yx

.How

ever

,fin

din

gso

luti

ons

of x

y�

yxan

d dr

awin

g it

s gr

aph

is

not

a s

impl

e pr

oces

s.

Sol

ve e

ach

pro

ble

m.A

ssu

me

that

xan

d y

are

pos

itiv

e re

al n

um

ber

s.

1.If

a�

0,w

ill

(a,a

) be

a s

olu

tion

of

xy�

yx?

Just

ify

you

r an

swer

.

Yes,

sin

ce a

a�

aam

ust

be

tru

e (R

efle

xive

Pro

p.o

f E

qu

alit

y).

2.If

c�

0,d

�0,

and

(c,d

) is

a s

olu

tion

of

xy�

yx,w

ill

(d,c

) al

so

be a

sol

uti

on?

Just

ify

you

r an

swer

.

Yes;

rep

laci

ng

xw

ith

d,y

wit

h c

giv

es d

c�

cd;

but

if (

c,d

) is

a s

olu

tio

n,

cd

�d

c .S

o,b

y th

e S

ymm

etri

c P

rop

erty

of

Eq

ual

ity,

dc

�c

dis

tru

e.

3.U

se 2

as

a va

lue

for

yin

xy

�yx

.Th

e eq

uat

ion

bec

omes

x2

�2x

.

a.F

ind

equ

atio

ns

for

two

fun

ctio

ns,

f(x)

an

d g(

x) t

hat

you

cou

ld g

raph

to

fin

d th

e so

luti

ons

of x

2�

2x.T

hen

gra

ph t

he

fun

ctio

ns

on a

sep

arat

esh

eet

of g

raph

pap

er.

f(x)

�x

2 ,g

(x)

�2

x

See

stu

den

ts’g

rap

hs.

b.

Use

th

e gr

aph

you

dre

w f

or p

art

ato

sta

te t

wo

solu

tion

s fo

r x2

�2x

.T

hen

use

th

ese

solu

tion

s to

sta

te t

wo

solu

tion

s fo

r xy

�yx

.2,

4;(2

,2),

(4,2

)

4.In

th

is e

xerc

ise,

a gr

aph

ing

calc

ula

tor

wil

l be

ver

y h

elpf

ul.

Use

th

e te

chn

iqu

e of

Exe

rcis

e 3

to c

ompl

ete

the

tabl

es b

elow

.Th

en g

raph

xy

�yx

for

posi

tive

val

ues

of

xan

d y.

If t

her

e ar

e as

ympt

otes

,sh

ow t

hem

in

you

rdi

agra

m u

sin

g do

tted

lin

es.N

ote

that

in

th

e ta

ble,

som

e va

lues

of

yca

llfo

r on

e va

lue

of x

,oth

ers

call

for

tw

o.

x

y

O

xy

44

24

55

1.8

5

88

1.5

8

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1 x

y

�1 2��1 2�

�3 4��3 4�

11

22

42

33

2.5

3


A


Stu

dy G

uid

e a

nd I

nte

rven

tion

Lo

gar

ith

ms

and

Lo

gar

ith

mic

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

©G

lenc

oe/M

cGra

w-H

ill57

9G

lenc

oe A

lgeb

ra 2

Lesson 10-2

Log

arit

hm

ic F

un

ctio

ns

and

Exp

ress

ion

s

Def

init

ion

of

Lo

gar

ith

m

Let

ban

d x

be p

ositi

ve n

umbe

rs,

b�

1. T

he lo

garit

hm o

f x

with

bas

e b

is d

enot

ed

wit

h B

ase

blo

g bx

and

is d

efin

ed a

s th

e ex

pone

nt y

that

mak

es t

he e

quat

ion

by

�x

true

.

Th

e in

vers

e of

th

e ex

pon

enti

al f

un

ctio

n y

�bx

is t

he

loga

rith

mic

fu

nct

ion

x�

by.

Th

is f

un

ctio

n i

s u

sual

ly w

ritt

en a

s y

�lo

g bx.

1.T

he f

unct

ion

is c

ontin

uous

and

one

-to-

one.

Pro

per

ties

of

2.T

he d

omai

n is

the

set

of

all p

ositi

ve r

eal n

umbe

rs.

Lo

gar

ith

mic

Fu

nct

ion

s3.

The

y-a

xis

is a

n as

ympt

ote

of t

he g

raph

.4.

The

ran

ge is

the

set

of

all r

eal n

umbe

rs.

5.T

he g

raph

con

tain

s th

e po

int

(0,

1).

Wri

te a

n e

xpon

enti

al e

qu

atio

n e

qu

ival

ent

to l

og3

243

�5.

35�

243

Wri

te a

log

arit

hm

ic e

qu

atio

n e

qu

ival

ent

to 6

�3

�.

log 6

��

3

Eva

luat

e lo

g 816

.

8�4 3�

�16

,so

log 8

16 �

.

Wri

te e

ach

eq

uat

ion

in

log

arit

hm

ic f

orm

.

1.27

�12

82.

3�4

�3.

��3

�

log

212

8 �

7lo

g3

��

4lo

g�1 7�

�3

Wri

te e

ach

eq

uat

ion

in

exp

onen

tial

for

m.

4.lo

g 15

225

�2

5.lo

g 3�

�3

6.lo

g 432

�

152

�22

53�

3�

4�5 2��

32

Eva

luat

e ea

ch e

xpre

ssio

n.

7.lo

g 464

38.

log 2

646

9.lo

g 100

100,

000

2.5

10.l

og5

625

411

.log

2781

12.l

og25

5

13.l

og2

�7

14.l

og10

0.00

001

�5

15.l

og4

�2.

51 � 32

1� 12

8

1 � 24 � 31 � 27

5 � 21 � 27

1� 34

31 � 81

1� 34

31 � 7

1 � 81

4 � 3

1� 21

6

1� 21

6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exam

ple3

Exam

ple3

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill58

0G

lenc

oe A

lgeb

ra 2

Solv

e Lo

gar

ith

mic

Eq

uat

ion

s an

d In

equ

alit

ies

Lo

gar

ith

mic

to

If

b�

1, x

�0,

and

log b

x�

y, t

hen

x�

by .

Exp

on

enti

al In

equ

alit

yIf

b�

1, x

�0,

and

log b

x�

y, t

hen

0 �

x�

by.

Pro

per

ty o

f E

qu

alit

y fo

r If

bis

a p

ositi

ve n

umbe

r ot

her

than

1,

Lo

gar

ith

mic

Fu

nct

ion

sth

en lo

g bx

�lo

g by

if an

d on

ly if

x�

y.

Pro

per

ty o

f In

equ

alit

y fo

r If

b�

1, t

hen

log b

x�

log b

yif

and

only

if x

�y,

L

og

arit

hm

ic F

un

ctio

ns

and

log b

x�

log b

yif

and

only

if x

�y.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Lo

gar

ith

ms

and

Lo

gar

ith

mic

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

Sol

ve l

og2

2x�

3.lo

g 22x

�3

Orig

inal

equ

atio

n

2x�

23D

efin

ition

of

loga

rithm

2x�

8S

impl

ify.

x�

4S

impl

ify.

Th

e so

luti

on i

s x

�4.

Sol

ve l

og5

(4x

�3)

�3.

log 5

(4x

�3)

�3

Orig

inal

equ

atio

n

0 �

4x�

3 �

53Lo

garit

hmic

to

expo

nent

ial i

nequ

ality

3 �

4x�

125

3

Add

ition

Pro

pert

y of

Ine

qual

ities

�x

�32

Sim

plify

.

Th

e so

luti

on s

et i

s �x |

�x

�32

.3 � 4

3 � 4

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

1.lo

g 232

�3x

2.lo

g 32c

��

2

3.lo

g 2x

16 �

�2

4.lo

g 25�

��10

5.lo

g 4(5

x

1) �

23

6.lo

g 8(x

�5)

�9

7.lo

g 4(3

x�

1) �

log 4

(2x

3)

48.

log 2

(x2

�6)

�lo

g 2(2

x

2)4

9.lo

g x

427

�3

�1

10.l

og2

(x

3) �

413

11.l

ogx

1000

�3

1012

.log

8(4

x

4) �

215

13.l

og2

2x�

2x

�2

14.l

og5

x�

2x

�25

15.l

og2

(3x

1)

�4

��

x�

516

.log

4(2

x) �

�x

�

17.l

og3

(x

3) �

3�

3 �

x�

2418

.log

276x

�x

�3 � 2

2 � 3

1 � 41 � 2

1 � 3

2 � 3

1 � 2x � 2

1 � 8

1 � 185 � 3



Skil

ls P

ract

ice

Lo

gar

ith

ms

and

Lo

gar

ith

mic

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

©G

lenc

oe/M

cGra

w-H

ill58

1G

lenc

oe A

lgeb

ra 2

Lesson 10-2

Wri

te e

ach

eq

uat

ion

in

log

arit

hm

ic f

orm

.

1.23

�8

log

28

�3

2.32

�9

log

39

�2

3.8�

2�

log

8�

�2

4.�

�2�

log

�1 3��

2

Wri

te e

ach

eq

uat

ion

in

exp

onen

tial

for

m.

5.lo

g 324

3 �

535

�24

36.

log 4

64 �

343

�64

7.lo

g 93

�9�1 2�

�3

8.lo

g 5�

�2

5�2

�

Eva

luat

e ea

ch e

xpre

ssio

n.

9.lo

g 525

210

.log

93

11.l

og10

1000

312

.log

125

5

13.l

og4

�3

14.l

og5

�4

15.l

og8

833

16.l

og27

�

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

ons.

17.l

og3

x�

524

318

.log

2x

�3

8

19.l

og4

y�

00

�y

�1

20.l

og�1 4�

x�

3

21.l

og2

n�

�2

n�

22.l

ogb

3 �

9

23.l

og6

(4x

12

) �

26

24.l

og2

(4x

�4)

�5

x�

9

25.l

og3

(x

2) �

log 3

(3x)

126

.log

6(3

y�

5)

log 6

(2y

3)

y

8

1 � 21 � 4

1 � 641 � 31 � 31

� 625

1 � 64

1 � 3

1 � 2

1 � 251 � 25

1 � 2

1 � 91 � 9

1 � 31 � 64

1 � 64

©G

lenc

oe/M

cGra

w-H

ill58

2G

lenc

oe A

lgeb

ra 2

Wri

te e

ach

eq

uat

ion

in

log

arit

hm

ic f

orm

.

1.53

�12

5lo

g5

125

�3

2.70

�1

log

71

�0

3.34

�81

log

381

�4

4.3�

4�

5.�

�3�

6.77

76�1 5�

�6

log

3�

�4

log

�1 4��

3lo

g77

766

�

Wri

te e

ach

eq

uat

ion

in

exp

onen

tial

for

m.

7.lo

g 621

6 �

363

�21

68.

log 2

64 �

626

�64

9.lo

g 3�

�4

3�4

�

10.l

og10

0.00

001

��

511

.log

255

�12

.log

328

�

10�

5�

0.00

001

25�1 2�

�5

32�3 5�

�8

Eva

luat

e ea

ch e

xpre

ssio

n.

13.l

og3

814

14.l

og10

0.00

01�

415

.log

2�

416

.log

�1 3�27

�3

17.l

og9

10

18.l

og8

419

.log

7�

220

.log

664

4

21.l

og3

�1

22.l

og4

�4

23.l

og9

9(n

1)

n�

124

.2lo

g 232

32

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

ons.

25.l

og10

n�

�3

26.l

og4

x�

3x

�64

27.l

og4

x�

8

28.l

og�1 5�

x�

�3

125

29.l

og7

q�

00

�q

�1

30.l

og6

(2y

8)

2

y

14

31.l

ogy

16 �

�4

32.l

ogn

��

32

33.l

ogb

1024

�5

4

34.l

og8

(3x

7)

�lo

g 8(7

x

4)35

.log

7(8

x

20)

�lo

g 7(x

6)

36.l

og3

(x2

�2)

�lo

g 3x

x�

�2

2

37.S

OU

ND

Sou

nds

th

at r

each

lev

els

of 1

30 d

ecib

els

or m

ore

are

pain

ful

to h

um

ans.

Wh

atis

th

e re

lati

ve i

nte

nsi

ty o

f 13

0 de

cibe

ls?

1013

38.I

NV

ESTI

NG

Mar

ia i

nve

sts

$100

0 in

a s

avin

gs a

ccou

nt

that

pay

s 8%

in

tere

stco

mpo

un

ded

ann

ual

ly.T

he

valu

e of

th

e ac

cou

nt

Aat

th

e en

d of

fiv

e ye

ars

can

be

dete

rmin

ed f

rom

th

e eq

uat

ion

log

A�

log[

1000

(1

0.08

)5].

Fin

d th

e va

lue

of A

to t

he

nea

rest

dol

lar.

$146

9

3 � 4

1 � 81 � 2

3 � 21

� 1000

1� 25

61 � 3

1 � 492 � 3

1 � 16

3 � 51 � 2

1 � 811 � 81

1 � 51 � 64

1 � 81

1 � 641 � 4

1 � 81Pra

ctic

e (

Ave

rag

e)

Lo

gar

ith

ms

and

Lo

gar

ith

mic

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2


A


Readin

g t

o L

earn

Math

em

ati

csL

og

arit

hm

s an

d L

og

arit

hm

ic F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

©G

lenc

oe/M

cGra

w-H

ill58

3G

lenc

oe A

lgeb

ra 2

Lesson 10-2

Pre-

Act

ivit

yW

hy

is a

log

arit

hm

ic s

cale

use

d t

o m

easu

re s

oun

d?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-2 a

t th

e to

p of

pag

e 53

1 in

you

r te

xtbo

ok.

How

man

y ti

mes

lou

der

than

a w

his

per

is n

orm

al c

onve

rsat

ion

?10

4o

r 10

,000

tim

es

Rea

din

g t

he

Less

on

1.a.

Wri

te a

n e

xpon

enti

al e

quat

ion

th

at i

s eq

uiv

alen

t to

log

381

�4.

34�

81

b.

Wri

te a

log

arit

hm

ic e

quat

ion

th

at i

s eq

uiv

alen

t to

25�

�1 2��

.lo

g25

��

c.W

rite

an

exp

onen

tial

equ

atio

n t

hat

is

equ

ival

ent

to l

og4

1 �

0.40

�1

d.

Wri

te a

log

arit

hm

ic e

quat

ion

th

at i

s eq

uiv

alen

t to

10�

3�

0.00

1.lo

g10

0.00

1 �

�3

e.W

hat

is

the

inve

rse

of t

he

fun

ctio

n y

�5x

?y

�lo

g5

x

f.W

hat

is

the

inve

rse

of t

he

fun

ctio

n y

�lo

g 10

x?y

�10

x

2.M

atch

eac

h f

un

ctio

n w

ith

its

gra

ph.

a.y

�3x

IVb

.y�

log 3

xI

c.y

��

�xII

I.II

.II

I.

3.In

dica

te w

het

her

eac

h o

f th

e fo

llow

ing

stat

emen

ts a

bou

t th

e ex

pon

enti

al f

un

ctio

n

y�

log 5

xis

tru

eor

fal

se.

a.T

he

y-ax

is i

s an

asy

mpt

ote

of t

he

grap

h.

tru

eb

.T

he

dom

ain

is

the

set

of a

ll r

eal

nu

mbe

rs.

fals

ec.

Th

e gr

aph

con

tain

s th

e po

int

(5,0

).fa

lse

d.

Th

e ra

nge

is

the

set

of a

ll r

eal

nu

mbe

rs.

tru

ee.

Th

e y-

inte

rcep

t is

1.

fals

e

Hel

pin

g Y

ou

Rem

emb

er4.

An

im

port

ant

skil

l n

eede

d fo

r w

orki

ng

wit

h l

ogar

ith

ms

is c

han

gin

g an

equ

atio

n b

etw

een

loga

rith

mic

and

exp

onen

tial

for

ms.

Usi

ng t

he w

ords

bas

e,ex

pone

nt,a

nd l

ogar

ithm

,des

crib

ean

eas

y w

ay t

o re

mem

ber

and

appl

y th

e pa

rt o

f th

e de

fin

itio

n o

f lo

gari

thm

th

at s

ays,

“log

bx

�y

if a

nd

only

if

by

�x.

”S

amp

le a

nsw

er:

In t

hes

e eq

uat

ion

s,b

stan

ds

for

bas

e.In

log

fo

rm,b

is t

he

sub

scri

pt,

and

in e

xpo

nen

tial

fo

rm,b

is t

he

num

ber

th

at is

rai

sed

to

a p

ower

.A lo

gar

ithm

is a

n e

xpo

nen

t,so

y,w

hic

h is

the

log

in t

he

first

eq

uat

ion

,bec

om

es t

he

exp

on

ent

in t

he

seco

nd

eq

uat

ion

.

x

y

Ox

y

O

x

y

O

1 � 3

1 � 21 � 5

1 � 5

©G

lenc

oe/M

cGra

w-H

ill58

4G

lenc

oe A

lgeb

ra 2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

Mu

sica

l Rel

atio

nsh

ips

Th

e fr

equ

enci

es o

f n

otes

in

a m

usi

cal

scal

e th

at a

re o

ne

octa

ve a

part

are

rela

ted

by a

n e

xpon

enti

al e

quat

ion

.For

th

e ei

ght

C n

otes

on

a p

ian

o,th

eeq

uat

ion

is

Cn

�C

12n

�1 ,

wh

ere

Cn

repr

esen

ts t

he

freq

uen

cy o

f n

ote

Cn.

1.F

ind

the

rela

tion

ship

bet

wee

n C

1an

d C

2.C

2�

2C1

2.F

ind

the

rela

tion

ship

bet

wee

n C

1an

d C

4.C

4�

8C1

Th

e fr

equ

enci

es o

f co

nse

cuti

ve n

otes

are

rel

ated

by

a

com

mon

rat

io r

.Th

e ge

ner

al e

quat

ion

is

f n�

f 1rn

�1 .

3.If

th

e fr

equ

ency

of

mid

dle

C i

s 26

1.6

cycl

es p

er s

econ

d an

d th

e fr

equ

ency

of

the

nex

t h

igh

er C

is

523.

2 cy

cles

pe

r se

con

d,fi

nd

the

com

mon

rat

io r

.(H

int:

Th

e tw

o C

’s

are

12 n

otes

apa

rt.)

Wri

te t

he

answ

er a

s a

radi

cal

expr

essi

on.

r�

12 �2�

4.S

ubs

titu

te d

ecim

al v

alu

es f

or r

and

f 1to

fin

d a

spec

ific

eq

uat

ion

for

fn.

f n�

261.

1(1.

0594

6)n

�1

5.F

ind

the

freq

uen

cy o

f F

#ab

ove

mid

dle

C.

f 7�

261.

6(1.

0594

6)6

�36

9.95

6.F

rets

are

a s

erie

s of

rid

ges

plac

ed a

cros

s th

e fi

nge

rboa

rd o

f a

guit

ar.T

hey

are

spac

ed s

o th

at t

he

sou

nd

mad

e by

pre

ssin

g a

stri

ng

agai

nst

on

e fr

eth

as a

bou

t 1.

0595

tim

es t

he

wav

elen

gth

of

the

sou

nd

mad

e by

usi

ng

the

nex

t fr

et.T

he

gen

eral

equ

atio

n i

s w

n�

w0(

1.05

95)n

.Des

crib

e th

ear

ran

gem

ent

of t

he

fret

s on

a g

uit

ar.

Th

e fr

ets

are

spac

ed in

a

log

arit

hm

ic s

cale

.



Stu

dy G

uid

e a

nd I

nte

rven

tion

Pro

per

ties

of

Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-3

10-3

©G

lenc

oe/M

cGra

w-H

ill58

5G

lenc

oe A

lgeb

ra 2

Lesson 10-3

Pro

per

ties

of

Log

arit

hm

sP

rope

rtie

s of

exp

onen

ts c

an b

e u

sed

to d

evel

op t

he

foll

owin

g pr

oper

ties

of

loga

rith

ms.

Pro

du

ct P

rop

erty

F

or a

ll po

sitiv

e nu

mbe

rs m

, n,

and

b,

whe

re b

�1,

o

f L

og

arit

hm

slo

g bm

n�

log b

m

log b

n.

Qu

oti

ent

Pro

per

ty

For

all

posi

tive

num

bers

m,

n, a

nd b

, w

here

b�

1,

of

Lo

gar

ith

ms

log b

�m n��

log b

m�

log b

n.

Po

wer

Pro

per

ty

For

any

rea

l num

ber

pan

d po

sitiv

e nu

mbe

rs m

and

b,

of

Lo

gar

ith

ms

whe

re b

�1,

log b

mp

�p

log b

m.

Use

log

328

3.

0331

an

d l

og3

4

1.26

19 t

o ap

pro

xim

ate

the

valu

e of

eac

h e

xpre

ssio

n.

Exam

ple

Exam

ple

a.lo

g 336

log 3

36�

log 3

(32

�4)

�lo

g 332

lo

g 34

�2

lo

g 34

2

1.

2619

3.

2619

b.

log 3

7

log 3

7�

log 3

��

�lo

g 328

�lo

g 34

3.

0331

�1.

2619

1.

7712

c.lo

g 325

6

log 3

256

�lo

g 3(4

4 )�

4 �

log 3

4

4(1.

2619

)

5.04

76

28 � 4

Exer

cises

Exer

cises

Use

log

123

0.

4421

an

d l

og12

7

0.78

31 t

o ev

alu

ate

each

exp

ress

ion

.

1.lo

g 12

211.

2252

2.lo

g 12

0.34

103.

log 1

249

1.56

62

4.lo

g 12

361.

4421

5.lo

g 12

631.

6673

6.lo

g 12

�0.

2399

7.lo

g 12

0.20

228.

log 1

216

,807

3.91

559.

log 1

244

12.

4504

Use

log

53

0.

6826

an

d l

og5

4

0.86

14 t

o ev

alu

ate

each

exp

ress

ion

.

10.l

og5

121.

5440

11.l

og5

100

2.86

1412

.log

50.

75�

0.17

88

13.l

og5

144

3.08

8014

.log

50.

3250

15.l

og5

375

3.68

26

16.l

og5

1.3�

0.17

8817

.log

5�

0.35

7618

.log

51.

7304

81 � 59 � 1627 � 16

81 � 49

27 � 49

7 � 3

©G

lenc

oe/M

cGra

w-H

ill58

6G

lenc

oe A

lgeb

ra 2

Solv

e Lo

gar

ith

mic

Eq

uat

ion

sYo

u c

an u

se t

he

prop

erti

es o

f lo

gari

thm

s to

sol

veeq

uat

ion

s in

volv

ing

loga

rith

ms.

Sol

ve e

ach

eq

uat

ion

.

a.2

log 3

x�

log 3

4 �

log 3

25

2 lo

g 3x

�lo

g 34

�lo

g 325

Orig

inal

equ

atio

n

log 3

x2�

log 3

4 �

log 3

25P

ower

Pro

pert

y

log 3

�lo

g 325

Quo

tient

Pro

pert

y

�25

Pro

pert

y of

Equ

ality

for

Log

arith

mic

Fun

ctio

ns

x2�

100

Mul

tiply

eac

h si

de b

y 4.

x�

�10

Take

the

squ

are

root

of

each

sid

e.

Sin

ce l

ogar

ith

ms

are

un

defi

ned

for

x�

0,�

10 i

s an

ext

ran

eou

s so

luti

on.

Th

e on

ly s

olu

tion

is

10.

b.

log 2

x

log 2

(x

2) �

3

log 2

x

log 2

(x

2) �

3O

rigin

al e

quat

ion

log 2

x(x

2)

�3

Pro

duct

Pro

pert

y

x(x

2)

�23

Def

initi

on o

f lo

garit

hm

x2

2x�

8D

istr

ibut

ive

Pro

pert

y

x2

2x �

8 �

0S

ubtr

act

8 fr

om e

ach

side

.

(x

4)(x

�2)

�0

Fac

tor.

x�

2or

x�

�4

Zer

o P

rodu

ct P

rope

rty

Sin

ce l

ogar

ith

ms

are

un

defi

ned

for

x�

0,�

4 is

an

ext

ran

eou

s so

luti

on.

Th

e on

ly s

olu

tion

is

2.

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

ons.

1.lo

g 54

lo

g 52x

�lo

g 524

32.

3 lo

g 46

�lo

g 48

�lo

g 4x

27

3.lo

g 625

lo

g 6x

�lo

g 620

44.

log 2

4 �

log 2

(x

3) �

log 2

8�

5.lo

g 62x

�lo

g 63

�lo

g 6(x

�1)

36.

2 lo

g 4(x

1)

�lo

g 4(1

1 �

x)2

7.lo

g 2x

�3

log 2

5 �

2 lo

g 210

12,5

008.

3 lo

g 2x

�2

log 2

5x�

210

0

9.lo

g 3(c

3)

�lo

g 3(4

c�

1) �

log 3

510

.log

5(x

3)

�lo

g 5(2

x�

1) �

24 � 7

8 � 19

5 � 21 � 2

x2� 4x2� 4

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Pro

per

ties

of

Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-3

10-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises


A


Skil

ls P

ract

ice

Pro

per

ties

of

Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-3

10-3

©G

lenc

oe/M

cGra

w-H

ill58

7G

lenc

oe A

lgeb

ra 2

Lesson 10-3

Use

log

23

1.

5850

an

d l

og2

5

2.32

19 t

o ap

pro

xim

ate

the

valu

e of

eac

hex

pre

ssio

n.

1.lo

g 225

4.64

382.

log 2

274.

755

3.lo

g 2�

0.73

694.

log 2

0.73

69

5.lo

g 215

3.90

696.

log 2

455.

4919

7.lo

g 275

6.22

888.

log 2

0.6

�0.

7369

9.lo

g 2�

1.58

5010

.log

20.

8481

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

ons.

11.l

og10

27 �

3 lo

g 10

x3

12.3

log

74

�2

log 7

b8

13.l

og4

5

log 4

x�

log 4

6012

14.l

og6

2c

log 6

8 �

log 6

805

15.l

og5

y�

log 5

8 �

log 5

18

16.l

og2

q�

log 2

3 �

log 2

721

17.l

og9

4

2 lo

g 95

�lo

g 9w

100

18.3

log

82

�lo

g 84

�lo

g 8b

2

19.l

og10

x

log 1

0(3

x�

5) �

log 1

02

220

.log

4x

lo

g 4(2

x�

3) �

log 4

22

21.l

og3

d

log 3

3 �

39

22.l

og10

y�

log 1

0(2

�y)

�0

1

23.l

og2

s

2 lo

g 25

�0

24.l

og2

(x

4) �

log 2

(x�

3) �

34

25.l

og4

(n

1) �

log 4

(n�

2) �

13

26.l

og5

10

log 5

12 �

3 lo

g 52

lo

g 5a

15

1 � 25

9 � 51 � 3

5 � 33 � 5

©G

lenc

oe/M

cGra

w-H

ill58

8G

lenc

oe A

lgeb

ra 2

Use

log

105

0.

6990

an

d l

og10

7

0.84

51 t

o ap

pro

xim

ate

the

valu

e of

eac

hex

pre

ssio

n.

1.lo

g 10

351.

5441

2.lo

g 10

251.

3980

3.lo

g 10

0.14

614.

log 1

0�

0.14

61

5.lo

g 10

245

2.38

926.

log 1

017

52.

2431

7.lo

g 10

0.2

�0.

6990

8.lo

g 10

0.55

29

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

ons.

9.lo

g 7n

�lo

g 78

410

.log

10u

�lo

g 10

48

11.l

og6

x

log 6

9 �

log 6

546

12.l

og8

48 �

log 8

w�

log 8

412

13.l

og9

(3u

14

) �

log 9

5 �

log 9

2u2

14.4

log

2x

lo

g 25

�lo

g 240

53

15.l

og3

y�

�lo

g 316

lo

g 364

16.l

og2

d�

5 lo

g 22

�lo

g 28

4

17.l

og10

(3m

�5)

lo

g 10

m�

log 1

02

218

.log

10(b

3)

lo

g 10

b�

log 1

04

1

19.l

og8

(t

10)

�lo

g 8(t

�1)

�lo

g 812

220

.log

3(a

3)

lo

g 3(a

2)

�lo

g 36

0

21.l

og10

(r

4) �

log 1

0r

�lo

g 10

(r

1)2

22.l

og4

(x2

�4)

�lo

g 4(x

2)

�lo

g 41

3

23.l

og10

4

log 1

0w

�2

2524

.log

8(n

�3)

lo

g 8(n

4)

�1

4

25.3

log

5(x

2

9) �

6 �

0�

426

.log

16(9

x

5) �

log 1

6(x

2�

1) �

3

27.l

og6

(2x

�5)

1

�lo

g 6(7

x

10)

828

.log

2(5

y

2) �

1 �

log 2

(1 �

2y)

0

29.l

og10

(c2

�1)

�2

�lo

g 10

(c

1)10

130

.log

7x

2

log 7

x�

log 7

3 �

log 7

726

31.S

OU

ND

Th

e lo

udn

ess

Lof

a s

oun

d in

dec

ibel

s is

giv

en b

y L

�10

log

10R

,wh

ere

Ris

th

eso

un

d’s

rela

tive

in

ten

sity

.If

the

inte

nsi

ty o

f a

cert

ain

sou

nd

is t

ripl

ed,b

y h

ow m

any

deci

bels

doe

s th

e so

un

d in

crea

se?

abo

ut

4.8

db

32.E

AR

THQ

UA

KES

An

ear

thqu

ake

rate

d at

3.5

on

th

e R

ich

ter

scal

e is

fel

t by

man

y pe

ople

,an

d an

ear

thqu

ake

rate

d at

4.5

may

cau

se l

ocal

dam

age.

Th

e R

ich

ter

scal

e m

agn

itu

dere

adin

g m

is g

iven

by

m�

log 1

0x,

wh

ere

xre

pres

ents

th

e am

plit

ude

of

the

seis

mic

wav

eca

usi

ng

grou

nd

mot

ion

.How

man

y ti

mes

gre

ater

is

the

ampl

itu

de o

f an

ear

thqu

ake

that

mea

sure

s 4.

5 on

th

e R

ich

ter

scal

e th

an o

ne

that

mea

sure

s 3.

5?10

tim

es

1 � 2

1 � 41 � 3

3 � 22 � 3

25 � 75 � 77 � 5

Pra

ctic

e (

Ave

rag

e)

Pro

per

ties

of

Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-3

10-3



Readin

g t

o L

earn

Math

em

ati

csP

rop

erti

es o

f L

og

arit

hm

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-3

10-3

©G

lenc

oe/M

cGra

w-H

ill58

9G

lenc

oe A

lgeb

ra 2

Lesson 10-3

Pre-

Act

ivit

yH

ow a

re t

he

pro

per

ties

of

exp

onen

ts a

nd

log

arit

hm

s re

late

d?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-3 a

t th

e to

p of

pag

e 54

1 in

you

r te

xtbo

ok.

Fin

d th

e va

lue

of l

og5

125.

3F

ind

the

valu

e of

log

55.

1F

ind

the

valu

e of

log

5(1

25

5).

2W

hic

h o

f th

e fo

llow

ing

stat

emen

ts i

s tr

ue?

BA

.lo

g 5(1

25

5) �

(log

512

5)

(log

55)

B.l

og5

(125

5)

�lo

g 512

5 �

log 5

5

Rea

din

g t

he

Less

on

1.E

ach

of

the

prop

erti

es o

f lo

gari

thm

s ca

n b

e st

ated

in

wor

ds o

r in

sym

bols

.Com

plet

e th

est

atem

ents

of

thes

e pr

oper

ties

in

wor

ds.

a.T

he

loga

rith

m o

f a

quot

ien

t is

th

e of

th

e lo

gari

thm

s of

th

e

and

the

.

b.

Th

e lo

gari

thm

of

a po

wer

is

the

of t

he

loga

rith

m o

f th

e ba

se a

nd

the

.

c.T

he

loga

rith

m o

f a

prod

uct

is

the

of t

he

loga

rith

ms

of i

ts

.

2.S

tate

wh

eth

er e

ach

of

the

foll

owin

g eq

uat

ion

s is

tru

eor

fal

se.I

f th

e st

atem

ent

is t

rue,

nam

e th

e pr

oper

ty o

f lo

gari

thm

s th

at i

s il

lust

rate

d.

a.lo

g 310

�lo

g 330

�lo

g 33

tru

e;Q

uo

tien

t P

rop

erty

b.l

og4

12 �

log 4

4

log 4

8fa

lse

c.lo

g 281

�2

log 2

9tr

ue;

Po

wer

Pro

per

tyd

.log

830

�lo

g 85

�lo

g 86

fals

e

3.T

he

alge

brai

c pr

oces

s of

sol

vin

g th

e eq

uat

ion

log

2x

lo

g 2(x

2)

�3

lead

s to

“x

��

4or

x�

2.”

Doe

s th

is m

ean

th

at b

oth

�4

and

2 ar

e so

luti

ons

of t

he

loga

rith

mic

equ

atio

n?

Exp

lain

you

r re

ason

ing.

Sam

ple

an

swer

:N

o;

2 is

a s

olu

tio

n b

ecau

se it

ch

ecks

:lo

g2

2 �

log

2(2

�2)

�lo

g2

2 �

log

24

�1

�2

�3.

Ho

wev

er,

bec

ause

log

2(�

4) a

nd

log

2(�

2) a

re u

nd

efin

ed,�

4 is

an

ext

ran

eou

sso

luti

on

an

d m

ust

be

elim

inat

ed.T

he

on

ly s

olu

tio

n is

2.

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r so

met

hing

is

to r

elat

e it

som

ethi

ng y

ou a

lrea

dy k

now

.Use

wor

dsto

exp

lain

how

th

e P

rodu

ct P

rope

rty

for

expo

nen

ts c

an h

elp

you

rem

embe

r th

e pr

odu

ctpr

oper

ty f

or l

ogar

ith

ms.

Sam

ple

an

swer

:Wh

en y

ou

mu

ltip

lytw

o n

um

ber

s o

rex

pre

ssio

ns

wit

h t

he

sam

e b

ase,

you

ad

dth

e ex

po

nen

ts a

nd

kee

p t

he

sam

e b

ase.

Lo

gar

ith

ms

are

exp

on

ents

,so

to

fin

d t

he

log

arit

hm

of

ap

rod

uct

,yo

u a

dd

the

log

arit

hm

s o

f th

e fa

cto

rs,k

eep

ing

th

e sa

me

bas

e.

fact

ors

sum

exp

on

ent

pro

du

ctd

eno

min

ato

rn

um

erat

or

dif

fere

nce

©G

lenc

oe/M

cGra

w-H

ill59

0G

lenc

oe A

lgeb

ra 2

Sp

iral

sC

onsi

der

an a

ngl

e in

sta

nda

rd p

osit

ion

wit

h i

ts v

erte

x at

a p

oin

t O

call

ed t

he

pole

.Its

in

itia

l si

de i

s on

a c

oord

inat

ized

axi

s ca

lled

th

e po

lar

axis

.A p

oin

t P

on t

he

term

inal

sid

e of

th

e an

gle

is n

amed

by

the

pola

r co

ord

inat

es(r

,�),

wh

ere

ris

th

e di

rect

ed d

ista

nce

of

the

poin

t fr

om O

and

�is

th

e m

easu

re o

fth

e an

gle.

Gra

phs

in t

his

sys

tem

may

be

draw

n o

n p

olar

coo

rdin

ate

pape

rsu

ch a

s th

e ki

nd

show

n b

elow

.

1.U

se a

cal

cula

tor

to c

ompl

ete

the

tabl

e fo

r lo

g 2r

�� 12�

0�.

(Hin

t:T

o fi

nd

�on

a c

alcu

lato

r,pr

ess

120

r2

.)

2.P

lot

the

poin

ts f

oun

d in

Exe

rcis

e 1

on t

he

grid

abo

ve a

nd

con

nec

t to

fo

rm a

sm

ooth

cu

rve.

Th

is t

ype

of s

pira

l is

cal

led

a lo

gari

thm

ic s

pira

l be

cau

se t

he

angl

e m

easu

res

are

prop

orti

onal

to

the

loga

rith

ms

of t

he

radi

i.

r1

23

45

67

8

�0�

120�

190�

240�

279�

310�

337�

360�

)

LOG

�)

LO

G�

01020

30

40

5060

7080

9010

011

012

013

0

140

150

160

170

180

190 200 21

0 220 23

0

240

250

260

270

280

290

300

310

32033

0340350

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-3

10-3


A


Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

mm

on

Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-4

10-4

©G

lenc

oe/M

cGra

w-H

ill59

1G

lenc

oe A

lgeb

ra 2

Lesson 10-4

Co

mm

on

Lo

gar

ith

ms

Bas

e 10

log

arit

hm

s ar

e ca

lled

com

mon

log

arit

hm

s.T

he

expr

essi

on l

og10

xis

usu

ally

wri

tten

wit

hou

t th

e su

bscr

ipt

as l

og x

.Use

th

e ke

y on

you

r ca

lcu

lato

r to

eva

luat

e co

mm

on l

ogar

ith

ms.

Th

e re

lati

on b

etw

een

exp

onen

ts a

nd

loga

rith

ms

give

s th

e fo

llow

ing

iden

tity

.

Inve

rse

Pro

per

ty o

f L

og

arit

hm

s an

d E

xpo

nen

ts10

log

x�

x

Eva

luat

e lo

g 50

to

fou

r d

ecim

al p

lace

s.U

se t

he

LO

G k

ey o

n y

our

calc

ula

tor.

To

fou

r de

cim

al p

lace

s,lo

g 50

�1.

6990

.

Sol

ve 3

2x�

1�

12.

32x

1

�12

Orig

inal

equ

atio

n

log

32x

1

�lo

g 12

Pro

pert

y of

Equ

ality

for

Log

arith

ms

(2x

1)

log

3 �

log

12P

ower

Pro

pert

y of

Log

arith

ms

2x

1 �

Div

ide

each

sid

e by

log

3.

2x�

�1

Sub

trac

t 1

from

eac

h si

de.

x�

��

1 �M

ultip

ly e

ach

side

by

.

x

0.63

09

Use

a c

alcu

lato

r to

eva

luat

e ea

ch e

xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.lo

g 18

2.lo

g 39

3.lo

g 12

01.

2553

1.59

112.

0792

4.lo

g 5.

85.

log

42.3

6.lo

g 0.

003

0.76

341.

6263

�2.

5229

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Rou

nd

to

fou

r d

ecim

al p

lace

s.

7.43

x�

120.

5975

8.6x

2

�18

�0.

3869

9.54

x�

2�

120

1.24

3710

.73x

�1

21

{x|x

0.

8549

}

11.2

.4x

4

�30

�0.

1150

12.6

.52x

20

0{x

|x

1.41

53}

13.3

.64x

�1

�85

.41.

1180

14.2

x

5�

3x�

213

.966

6

15.9

3x�

45x

2

�8.

1595

16.6

x�

5�

27x

3

�3.

6069

1 � 2lo

g 12

� log

31 � 2lo

g 12

� log

3

log

12� lo

g 3

LOG

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

©G

lenc

oe/M

cGra

w-H

ill59

2G

lenc

oe A

lgeb

ra 2

Ch

ang

e o

f B

ase

Form

ula

Th

e fo

llow

ing

form

ula

is

use

d to

ch

ange

exp

ress

ion

s w

ith

diff

eren

t lo

gari

thm

ic b

ases

to

com

mon

log

arit

hm

exp

ress

ion

s.

Ch

ang

e o

f B

ase

Fo

rmu

laF

or a

ll po

sitiv

e nu

mbe

rs a

, b,

and

n,

whe

re a

�1

and

b�

1, lo

g an

�

Exp

ress

log

815

in

ter

ms

of c

omm

on l

ogar

ith

ms.

Th

en a

pp

roxi

mat

eit

s va

lue

to f

our

dec

imal

pla

ces.

log 8

15�

Cha

nge

of B

ase

For

mul

a

1.

3023

Sim

plify

.

Th

e va

lue

of l

og8

15 i

s ap

prox

imat

ely

1.30

23.

Exp

ress

eac

h l

ogar

ith

m i

n t

erm

s of

com

mon

log

arit

hm

s.T

hen

ap

pro

xim

ate

its

valu

e to

fou

r d

ecim

al p

lace

s.

1.lo

g 316

2.lo

g 240

3.lo

g 535

,2.5

237

,5.3

219

,2.2

091

4.lo

g 422

5.lo

g 12

200

6.lo

g 250

,2.2

297

,2.1

322

,5.6

439

7.lo

g 50.

48.

log 3

29.

log 4

28.5

,�0.

5693

,0.6

309

,2.4

164

10.l

og3

(20)

211

.log

6(5

)412

.log

8(4

)5

,5.4

537

,3.5

930

,3.3

333

13.l

og5

(8)3

14.l

og2

(3.6

)615

.log

12(1

0.5)

4

,3.8

761

,11.

0880

,3.7

851

16.l

og3

�15

0�

17.l

og4

3 �39�

18.l

og5

4 �16

00�

,2.2

804

,0.8

809

,1.1

460

log

160

0�

�4

log

5lo

g 3

9� 3

log

4lo

g 1

50� 2

log

3

4 lo

g 1

0.5

��

log

12

6 lo

g 3

.6�

�lo

g 2

3 lo

g 8

�lo

g 5

5 lo

g 4

�lo

g 8

4 lo

g 5

�lo

g 6

2 lo

g 2

0�

�lo

g 3

log

28.

5�

�lo

g 4

log

2� lo

g 3

log

0.4

�lo

g 5

log

50

� log

2lo

g 2

00� lo

g 1

2lo

g 2

2� lo

g 4

log

35

� log

5lo

g 4

0� lo

g 2

log

16

� log

3

log 10

15� lo

g 108

log b

n� lo

g ba

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Co

mm

on

Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-4

10-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Co

mm

on

Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-4

10-4

©G

lenc

oe/M

cGra

w-H

ill59

3G

lenc

oe A

lgeb

ra 2

Lesson 10-4

Use

a c

alcu

lato

r to

eva

luat

e ea

ch e

xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.lo

g 6

0.77

822.

log

151.

1761

3.lo

g 1.

10.

0414

4.lo

g 0.

3�

0.52

29

Use

th

e fo

rmu

la p

H �

�lo

g[H

�]

to f

ind

th

e p

H o

f ea

ch s

ub

stan

ce g

iven

its

con

cen

trat

ion

of

hyd

roge

n i

ons.

5.ga

stri

c ju

ices

:[H

]

�1.

0 �

10�

1m

ole

per

lite

r1.

0

6.to

mat

o ju

ice:

[H

] �

7.94

�10

�5

mol

e pe

r li

ter

4.1

7.bl

ood:

[H

] �

3.98

�10

�8

mol

e pe

r li

ter

7.4

8.to

oth

past

e:[H

]

�1.

26 �

10�

10m

ole

per

lite

r9.

9

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Rou

nd

to

fou

r d

ecim

al p

lace

s.

9.3x

�24

3{x

|x�

5}10

.16v

��v �

v

�

11.8

p�

501.

8813

12.7

y�

151.

3917

13.5

3b�

106

0.96

5914

.45k

�37

0.52

09

15.1

27p

�12

00.

2752

16.9

2m�

270.

75

17.3

r�

5�

4.1

6.28

4318

.8y

4

�15

{y|y

��

2.69

77}

19.7

.6d

3

�57

.2�

1.00

4820

.0.5

t�

8�

16.3

3.97

32

21.4

2x2

�84

�1.

0888

22.5

x2

1 �10

�0.

6563

Exp

ress

eac

h l

ogar

ith

m i

n t

erm

s of

com

mon

log

arit

hm

s.T

hen

ap

pro

xim

ate

its

valu

e to

fou

r d

ecim

al p

lace

s.

23.l

og3

7;

1.77

1224

.log

566

;2.

6032

25.l

og2

35;

5.12

9326

.log

610

;1.

2851

log

1010

��

log

106

log

1035

��

log

102

log

1066

��

log

105

log

107

� log

103

1 � 21 � 4

©G

lenc

oe/M

cGra

w-H

ill59

4G

lenc

oe A

lgeb

ra 2

Use

a c

alcu

lato

r to

eva

luat

e ea

ch e

xpre

ssio

n t

o fo

ur

dec

imal

pla

ces.

1.lo

g 10

12.

0043

2.lo

g 2.

20.

3424

3.lo

g 0.

05�

1.30

10

Use

th

e fo

rmu

la p

H �

�lo

g[H

�]

to f

ind

th

e p

H o

f ea

ch s

ub

stan

ce g

iven

its

con

cen

trat

ion

of

hyd

roge

n i

ons.

4.m

ilk:

[H

] �

2.51

�10

�7

mol

e pe

r li

ter

6.6

5.ac

id r

ain

:[H

]

�2.

51 �

10�

6m

ole

per

lite

r5.

6

6.bl

ack

coff

ee:[

H

] �

1.0

�10

�5

mol

e pe

r li

ter

5.0

7.m

ilk

of m

agn

esia

:[H

]

�3.

16 �

10�

11m

ole

per

lite

r10

.5

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Rou

nd

to

fou

r d

ecim

al p

lace

s.

8.2x

�25

{x|x

�4.

6439

}9.

5a�

120

2.97

4610

.6z

�45

.62.

1319

11.9

m

100

{m|m

2.

0959

}12

.3.5

x�

47.9

3.08

8513

.8.2

y�

64.5

1.98

02

14.2

b

1�

7.31

{b|b

1.

8699

}15.

42x

�27

1.18

8716

.2a

�4

�82

.110

.359

3

17.9

z�

2�

38{z

|z�

3.65

55}

18.5

w

3�

17�

1.23

9619

.30x

2�

50�

1.07

25

20.5

x2�

3�

72�

2.37

8521

.42x

�9x

1

3.81

8822

.2n

1

�52

n�

10.

9117

Exp

ress

eac

h l

ogar

ith

m i

n t

erm

s of

com

mon

log

arit

hm

s.T

hen

ap

pro

xim

ate

its

valu

e to

fou

r d

ecim

al p

lace

s.

23.l

og5

12;

1.54

4024

.log

832

;1.

6667

25.l

og11

9 ;

0.91

63

26.l

og2

18

;4.

1699

27.l

og9

6;

0.81

5528

.log

7�

8�;

29.H

OR

TIC

ULT

UR

ES

iber

ian

iri

ses

flou

rish

wh

en t

he

con

cen

trat

ion

of

hyd

roge

n i

ons

[H

]in

th

e so

il i

s n

ot l

ess

than

1.5

8 �

10�

8m

ole

per

lite

r.W

hat

is

the

pH o

f th

e so

il i

n w

hic

hth

ese

iris

es w

ill

flou

rish

?7.

8 o

r le

ss

30.A

CID

ITY

Th

e pH

of

vin

egar

is

2.9

and

the

pH o

f m

ilk

is 6

.6.H

ow m

any

tim

es g

reat

er i

sth

e h

ydro

gen

ion

con

cen

trat

ion

of

vin

egar

th

an o

f m

ilk?

abo

ut

5000

31.B

IOLO

GY

Th

ere

are

init

iall

y 10

00 b

acte

ria

in a

cu

ltu

re.T

he

nu

mbe

r of

bac

teri

a do

ubl

esea

ch h

our.

Th

e n

um

ber

of b

acte

ria

Npr

esen

t af

ter

th

ours

is

N�

1000

(2)t

.How

lon

g w

ill

it t

ake

the

cult

ure

to

incr

ease

to

50,0

00 b

acte

ria?

abo

ut

5.6

h

32.S

OU

ND

An

equ

atio

n f

or l

oudn

ess

Lin

dec

ibel

s is

giv

en b

y L

�10

log

R,w

her

e R

is t

he

sou

nd’

s re

lati

ve i

nte

nsi

ty.A

n a

ir-r

aid

sire

n c

an r

each

150

dec

ibel

s an

d je

t en

gin

e n

oise

can

rea

ch 1

20 d

ecib

els.

How

man

y ti

mes

gre

ater

is

the

rela

tive

in

ten

sity

of

the

air-

raid

sire

n t

han

th

at o

f th

e je

t en

gin

e n

oise

?10

00

log

108

� 2 lo

g10

7lo

g10

6�

�lo

g10

9lo

g10

18�

�lo

g10

2

log

109

��

log

1011

log

1032

��

log

108

log

1012

��

log

105

Pra

ctic

e (

Ave

rag

e)

Co

mm

on

Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-4

10-4

0.53

43


A


Readin

g t

o L

earn

Math

em

ati

csC

om

mo

n L

og

arit

hm

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-4

10-4

©G

lenc

oe/M

cGra

w-H

ill59

5G

lenc

oe A

lgeb

ra 2

Lesson 10-4

Pre-

Act

ivit

yW

hy

is a

log

arit

hm

ic s

cale

use

d t

o m

easu

re a

cid

ity?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-4 a

t th

e to

p of

pag

e 54

7 in

you

r te

xtbo

ok.

Wh

ich

su

bsta

nce

is

mor

e ac

idic

,mil

k or

tom

atoe

s?

tom

ato

es

Rea

din

g t

he

Less

on

1.R

hon

da u

sed

the

foll

owin

g ke

ystr

okes

to

ente

r an

exp

ress

ion

on

her

gra

phin

g ca

lcu

lato

r:

17

Th

e ca

lcu

lato

r re

turn

ed t

he

resu

lt 1

.230

4489

21.

Wh

ich

of

the

foll

owin

g co

ncl

usi

ons

are

corr

ect?

a,c,

and

d

a.T

he

base

10

loga

rith

m o

f 17

is

abou

t 1.

2304

.

b.

Th

e ba

se 1

7 lo

gari

thm

of

10 i

s ab

out

1.23

04.

c.T

he

com

mon

log

arit

hm

of

17 i

s ab

out

1.23

0449

.

d.

101.

2304

4892

1is

ver

y cl

ose

to 1

7.

e.T

he

com

mon

log

arit

hm

of

17 i

s ex

actl

y 1.

2304

4892

1.

2.M

atch

eac

h e

xpre

ssio

n f

rom

th

e fi

rst

colu

mn

wit

h a

n e

xpre

ssio

n f

rom

th

e se

con

d co

lum

nth

at h

as t

he

sam

e va

lue.

a.lo

g 22

ivi.

log 4

1

b.

log

12 ii

iii

.log

28

c.lo

g 31

iii

i.lo

g 10

12

d.

log 5

viv

.log

55

e.lo

g 10

00ii

v.lo

g 0.

1

3.C

alcu

lato

rs d

o n

ot h

ave

keys

for

fin

din

g ba

se 8

log

arit

hm

s di

rect

ly.H

owev

er,y

ou c

an u

se

a ca

lcu

lato

r to

fin

d lo

g 820

if

you

app

ly t

he

form

ula

.

Wh

ich

of

the

foll

owin

g ex

pres

sion

s ar

e eq

ual

to

log 8

20?

B a

nd

C

A.l

og20

8B

.C

.D

.

Hel

pin

g Y

ou

Rem

emb

er

4.S

omet

imes

it

is e

asie

r to

rem

embe

r a

form

ula

if

you

can

sta

te i

t in

wor

ds.S

tate

th

ech

ange

of

base

for

mu

la i

n w

ords

.S

amp

le a

nsw

er:T

o c

han

ge

the

log

arit

hm

of

an

um

ber

fro

m o

ne

bas

e to

an

oth

er,d

ivid

e th

e lo

g o

f th

e o

rig

inal

nu

mb

erin

th

e o

ld b

ase

by t

he

log

of

the

new

bas

e in

th

e o

ld b

ase.

log

8� lo

g 20

log

20� lo

g 8

log 10

20� lo

g 108

chan

ge

of

bas

e

1 � 5

ENTE

R)

LO

G

©G

lenc

oe/M

cGra

w-H

ill59

6G

lenc

oe A

lgeb

ra 2

Th

e S

lide

Ru

leB

efor

e th

e in

ven

tion

of

elec

tron

ic c

alcu

lato

rs,c

ompu

tati

ons

wer

e of

ten

perf

orm

ed o

n a

sli

de r

ule

.A s

lide

ru

le i

s ba

sed

on t

he

idea

of

loga

rith

ms.

It h

astw

o m

ovab

le r

ods

labe

led

wit

h C

an

d D

sca

les.

Eac

h o

f th

e sc

ales

is

loga

rith

mic

.

To

mu

ltip

ly 2

�3

on a

sli

de r

ule

,mov

e th

e C

rod

to

the

righ

t as

sh

own

belo

w.Y

ou c

an f

ind

2 �

3 by

add

ing

log

2 to

log

3,a

nd

the

slid

e ru

le a

dds

the

len

gth

s fo

r yo

u.T

he

dist

ance

you

get

is

0.77

8,or

th

e lo

gari

thm

of

6.

Fol

low

th

e st

eps

to m

ake

a sl

ide

rule

.

1.U

se g

raph

pap

er t

hat

has

sm

all

squ

ares

,su

ch a

s 10

squ

ares

to

the

inch

.Usi

ng

the

scal

es s

how

n a

t th

e ri

ght,

plot

th

e cu

rve

y�

log

xfo

r x

�1,

1.5,

and

the

wh

ole

nu

mbe

rs f

rom

2 t

hro

ugh

10.

Mak

e an

obv

iou

s h

eavy

dot

for

eac

h p

oin

t pl

otte

d.

2.Yo

u w

ill

nee

d tw

o st

rips

of

card

boar

d.A

5-

by-7

in

dex

card

,cu

t in

hal

f th

e lo

ng

way

,w

ill

wor

k fi

ne.

Tu

rn t

he

grap

h y

ou m

ade

in

Exe

rcis

e 1

side

way

s an

d u

se i

t to

mar

ka

loga

rith

mic

sca

le o

n e

ach

of

the

two

stri

ps.T

he

figu

re s

how

s th

e m

ark

for

2 be

ing

draw

n.

3.E

xpla

in h

ow t

o u

se a

sli

de r

ule

to

divi

de 8

by

2.L

ine

up

th

e 2

on

th

e C

sca

le w

ith

th

e 8

on

th

e D

sca

le.T

he

qu

oti

ent

is t

he

nu

mb

er o

n t

he

D s

cale

bel

ow

th

e 1

on

th

e C

sca

le.

0

0.1

0.2

0.3

y

1 2

11.

52

y =

log

x

0.1

0.2

12

1 21

CD

2 4

3 6

45

67

89

83

57

9

log

6

log

3lo

g 2

12

34

56

78

9

12

34

56

78

9

C D

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-4

10-4

1–2.

See

st

ud

ents

’wo

rk.



Stu

dy G

uid

e a

nd I

nte

rven

tion

Bas

e e

and

Nat

ura

l Lo

gar

ith

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-5

10-5

©G

lenc

oe/M

cGra

w-H

ill59

7G

lenc

oe A

lgeb

ra 2

Lesson 10-5

Bas

e e

and

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ura

l Lo

gar

ith

ms

Th

e ir

rati

onal

nu

mbe

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2.

7182

8… o

ften

occ

urs

as t

he

base

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exp

onen

tial

an

d lo

gari

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ic f

un

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ns

that

des

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nin

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�1

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e

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.

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e fu

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and

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fun

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per

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f B

ase

ean

d N

atu

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arit

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sel

n x

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x

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se e

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can

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uat

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sin

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and

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our

calc

ula

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luat

e ln

168

5.U

se a

cal

cula

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ln 1

685

7.

4295

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.520

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wit

h a

n e

xke

y to

fin

d e

to 7

dec

imal

pla

ces.

2.71

8281

8

2.U

se t

he

expr

essi

on �1

� n1 � �n

to a

ppro

xim

ate

eto

3 d

ecim

al p

lace

s.U

se

5,10

0,50

0,an

d 70

00 a

s va

lues

of

n.

2.48

8,2.

705,

2.71

6,2.

718

3.U

se t

he

infi

nit

e su

m t

o ap

prox

imat

e e

to 3

dec

imal

pla

ces.

Use

th

e w

hol

e n

um

bers

fro

m 3

th

rou

gh 6

as

valu

es o

f n

.2.

667,

2.70

8,2.

717,

2.71

8

4.W

hic

h a

ppro

xim

atio

n m

eth

od a

ppro

ach

es t

he

valu

e of

em

ore

quic

kly?

the

infi

nit

e su

m5.

Use

a c

alcu

lato

r w

ith

a �

key

to fi

nd

�to

7 d

ecim

al p

lace

s.3.

1415

927

6.U

se t

he

infi

nit

e pr

odu

ct t

o ap

prox

imat

e �

to 3

dec

imal

pla

ces.

Use

th

e w

hol

e n

um

bers

fro

m 3

th

rou

gh 6

as

valu

es o

f n

.2.

926,

2.97

2,3.

002,

3.02

3

7.D

oes

the

infi

nit

e pr

odu

ct g

ive

good

app

roxi

mat

ion

s fo

r �

quic

kly?

no

8.S

how

th

at �

4

�5

is e

qual

to

e6to

4 d

ecim

al p

lace

s.To

4 d

ecim

al p

lace

s,th

ey b

oth

eq

ual

403

.428

8.9.

Wh

ich

is

larg

er,e

�or

�e ?

e�>

�e

10.T

he

expr

essi

on x

reac

hes

a m

axim

um

val

ue

at x

�e.

Use

th

is f

act

to

prov

e th

e in

equ

alit

y yo

u f

oun

d in

Exe

rcis

e 9.

e�1 e�>

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dy G

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e a

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nte

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Exp

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al G

row

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nd

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ay

NA

ME

____

____

____

____

____

____

____

____

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AT

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Exp

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ecay

Dep

reci

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f va

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and

radi

oact

ive

deca

y ar

e ex

ampl

es o

fex

pon

enti

al d

ecay

.Wh

en a

qu

anti

ty d

ecre

ases

by

a fi

xed

perc

ent

each

tim

e pe

riod

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eam

oun

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th

e qu

anti

ty a

fter

tti

me

peri

ods

is g

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by

y�

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t ,w

her

e a

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amou

nt

and

ris

th

e pe

rcen

t de

crea

se e

xpre

ssed

as

a de

cim

al.

An

oth

er e

xpon

enti

al d

ecay

mod

el o

ften

use

d by

sci

enti

sts

is y

�ae

�kt

,wh

ere

kis

a c

onst

ant.

CO

NSU

MER

PR

ICES

As

tech

nol

ogy

adva

nce

s,th

e p

rice

of

man

yte

chn

olog

ical

dev

ices

su

ch a

s sc

ien

tifi

c ca

lcu

lato

rs a

nd

cam

cord

ers

goes

dow

n.

On

e b

ran

d o

f h

and

-hel

d o

rgan

izer

sel

ls f

or $

89.

a.If

its

pri

ce d

ecre

ases

by

6% p

er y

ear,

how

mu

ch w

ill

it c

ost

afte

r 5

year

s?U

se t

he

expo

nen

tial

dec

ay m

odel

wit

h i

nit

ial

amou

nt

$89,

perc

ent

decr

ease

0.0

6,an

dti

me

5 ye

ars.

y�

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tE

xpon

entia

l dec

ay f

orm

ula

y�

89(1

�0.

06)5

a�

89,

r�

0.06

, t

�5

y�

$65.

32A

fter

5 y

ears

th

e pr

ice

wil

l be

$65

.32.

b.

Aft

er h

ow m

any

year

s w

ill

its

pri

ce b

e $5

0?To

fin

d w

hen

the

pric

e w

ill b

e $5

0,ag

ain

use

the

expo

nent

ial d

ecay

for

mul

a an

d so

lve

for

t.y

�a(

1 �

r)t

Exp

onen

tial d

ecay

for

mul

a

50 �

89(1

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y�

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a�

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r�

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�(0

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tD

ivid

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ch s

ide

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9.

log

��

log

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pert

y of

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ality

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arith

ms

log

��

tlo

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94P

ower

Pro

pert

y

t�

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ide

each

sid

e by

log

0.94

.

t

9.3

Th

e pr

ice

wil

l be

$50

aft

er a

bou

t 9.

3 ye

ars.

1.B

USI

NES

SA

fu

rnit

ure

sto

re i

s cl

osin

g ou

t it

s bu

sin

ess.

Eac

h w

eek

the

own

er l

ower

spr

ices

by

25%

.Aft

er h

ow m

any

wee

ks w

ill

the

sale

pri

ce o

f a

$500

ite

m d

rop

belo

w $

100?

6 w

eeks

CA

RB

ON

DA

TIN

GU

se t

he

form

ula

y�

ae�

0.00

012t

,wh

ere

ais

th

e in

itia

l am

oun

t of

Car

bon

-14,

tis

th

e n

um

ber

of

year

s ag

o th

e an

imal

liv

ed,a

nd

yis

th

e re

mai

nin

gam

oun

t af

ter

tye

ars.

2.H

ow o

ld is

a f

ossi

l rem

ain

that

has

lost

95%

of

its

Car

bon-

14?

abo

ut

25,0

00 y

ears

old

3.H

ow o

ld is

a s

kele

ton

that

has

95%

of

its

Car

bon-

14 r

emai

ning

?ab

ou

t 42

7.5

year

s o

ld

log

��5 80 9��

��

log

0.94

50 � 8950 � 8950 � 89

Exam

ple

Exam

ple

Exer

cises

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cises

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opu

lati

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an

d gr

owth

of

bact

eria

col

onie

s ar

e ex

ampl

esof

exp

onen

tial

gro

wth

.Wh

en a

qu

anti

ty i

ncr

ease

s by

a f

ixed

per

cen

t ea

ch t

ime

peri

od,t

he

amou

nt

of t

hat

qu

anti

ty a

fter

tti

me

peri

ods

is g

iven

by

y�

a(1

r)

t ,w

her

e a

is t

he

init

ial

amou

nt

and

ris

th

e pe

rcen

t in

crea

se (

or r

ate

of g

row

th)

expr

esse

d as

a d

ecim

al.

An

oth

er e

xpon

enti

al g

row

th m

odel

oft

en u

sed

by s

cien

tist

s is

y�

aekt

,wh

ere

kis

a c

onst

ant.

A c

omp

ute

r en

gin

eer

is h

ired

for

a s

alar

y of

$28

,000

.If

she

gets

a5%

rai

se e

ach

yea

r,af

ter

how

man

y ye

ars

wil

l sh

e b

e m

akin

g $5

0,00

0 or

mor

e?U

se t

he

expo

nen

tial

gro

wth

mod

el w

ith

a�

28,0

00,y

�50

,000

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d r

�0.

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nd

solv

e fo

r t.

y�

a(1

r)

tE

xpon

entia

l gro

wth

for

mul

a

50,0

00 �

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00(1

0.

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ivid

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ch s

ide

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pert

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ality

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rithm

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log

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tlo

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ower

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pert

y

t�

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ide

each

sid

e by

log

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t

11.9

yea

rsU

se a

cal

cula

tor.

If r

aise

s ar

e gi

ven

an

nu

ally

,sh

e w

ill

be m

akin

g ov

er $

50,0

00 i

n 1

2 ye

ars.

1.B

AC

TER

IA G

RO

WTH

A c

erta

in s

trai

n o

f ba

cter

ia g

row

s fr

om 4

0 to

326

in

120

min

ute

s.F

ind

kfo

r th

e gr

owth

for

mu

la y

�ae

kt,w

her

e t

is i

n m

inu

tes.

abo

ut

0.01

75

2.IN

VES

TMEN

TC

arl

plan

s to

in

vest

$50

0 at

8.2

5% i

nte

rest

,com

pou

nde

d co

nti

nu

ousl

y.H

ow l

ong

wil

l it

tak

e fo

r h

is m

oney

to

trip

le?

abo

ut

14 y

ears

3.SC

HO

OL

POPU

LATI

ON

Th

ere

are

curr

entl

y 85

0 st

ude

nts

at

the

hig

h s

choo

l,w

hic

hre

pres

ents

fu

ll c

apac

ity.

Th

e to

wn

pla

ns

an a

ddit

ion

to

hou

se 4

00 m

ore

stu

den

ts.I

f th

e sc

hoo

l po

pula

tion

gro

ws

at 7

.8%

per

yea

r,in

how

man

y ye

ars

wil

l th

e n

ew a

ddit

ion

be f

ull

?ab

ou

t 5

year

s

4.EX

ERC

ISE

Hu

go b

egin

s a

wal

kin

g pr

ogra

m b

y w

alki

ng

mil

e pe

r da

y fo

r on

e w

eek.

Eac

h w

eek

ther

eaft

er h

e in

crea

ses

his

mil

eage

by

10%

.Aft

er h

ow m

any

wee

ks i

s h

ew

alki

ng

mor

e th

an 5

mil

es p

er d

ay?

24 w

eeks

5.V

OC

AB

ULA

RY G

RO

WTH

Wh

en E

mil

y w

as 1

8 m

onth

s ol

d,sh

e h

ad a

10-

wor

dvo

cabu

lary

.By

the

tim

e sh

e w

as 5

yea

rs o

ld (

60 m

onth

s),h

er v

ocab

ular

y w

as 2

500

wor

ds.

If h

er v

ocab

ular

y in

crea

sed

at a

con

stan

t pe

rcen

t pe

r m

onth

,wha

t w

as t

hat

incr

ease

?ab

ou

t 14

%

1 � 2

log

��5 20 8��

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50 � 2850 � 2850 � 28

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Exp

on

enti

al G

row

th a

nd

Dec

ay

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-6

10-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Exp

on

enti

al G

row

th a

nd

Dec

ay

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

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____

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10-6

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Sol

ve e

ach

pro

ble

m.

1.FI

SHIN

GIn

an

ove

r-fi

shed

are

a,th

e ca

tch

of

a ce

rtai

n f

ish

is

decr

easi

ng

at a

n a

vera

gera

te o

f 8%

per

yea

r.If

th

is d

ecli

ne

pers

ists

,how

lon

g w

ill

it t

ake

for

the

catc

h t

o re

ach

hal

f of

th

e am

oun

t be

fore

th

e de

clin

e?ab

ou

t 8.

3 yr

2.IN

VES

TIN

GA

lex

inve

sts

$200

0 in

an

acc

oun

t th

at h

as a

6%

an

nu

al r

ate

of g

row

th.T

oth

e n

eare

st y

ear,

wh

en w

ill

the

inve

stm

ent

be w

orth

$36

00?

10 y

r

3.PO

PULA

TIO

NA

cu

rren

t ce

nsu

s sh

ows

that

th

e po

pula

tion

of

a ci

ty i

s 3.

5 m

illi

on.U

sin

gth

e fo

rmu

la P

�ae

rt,f

ind

the

expe

cted

pop

ula

tion

of

the

city

in

30

year

s if

th

e gr

owth

rate

rof

th

e po

pula

tion

is

1.5%

per

yea

r,a

repr

esen

ts t

he

curr

ent

popu

lati

on i

n m

illi

ons,

and

tre

pres

ents

th

e ti

me

in y

ears

.ab

ou

t 5.

5 m

illio

n

4.PO

PULA

TIO

NT

he

popu

lati

on P

in t

hou

san

ds o

f a

city

can

be

mod

eled

by

the

equ

atio

nP

�80

e0.0

15t ,

wh

ere

tis

th

e ti

me

in y

ears

.In

how

man

y ye

ars

wil

l th

e po

pula

tion

of

the

city

be

120,

000?

abo

ut

27 y

r

5.B

AC

TER

IAH

ow m

any

days

wil

l it

tak

e a

cult

ure

of

bact

eria

to

incr

ease

fro

m 2

000

to50

,000

if

the

grow

th r

ate

per

day

is 9

3.2%

?ab

ou

t 4.

9 d

ays

6.N

UC

LEA

R P

OW

ERT

he

elem

ent

plu

ton

ium

-239

is

hig

hly

rad

ioac

tive

.Nu

clea

r re

acto

rsca

n p

rodu

ce a

nd

also

use

th

is e

lem

ent.

Th

e h

eat

that

plu

ton

ium

-239

em

its

has

hel

ped

topo

wer

equ

ipm

ent

on t

he

moo

n.I

f th

e h

alf-

life

of

plu

ton

ium

-239

is

24,3

60 y

ears

,wh

at i

sth

e va

lue

of k

for

this

ele

men

t?ab

ou

t 0.

0000

2845

7.D

EPR

ECIA

TIO

NA

Glo

bal

Pos

itio

nin

g S

atel

lite

(G

PS

) sy

stem

use

s sa

tell

ite

info

rmat

ion

to l

ocat

e gr

oun

d po

siti

on.A

bu’s

su

rvey

ing

firm

bou

ght

a G

PS

sys

tem

for

$12

,500

.Th

eG

PS

dep

reci

ated

by

a fi

xed

rate

of

6% a

nd

is n

ow w

orth

$86

00.H

ow l

ong

ago

did

Abu

buy

the

GP

S s

yste

m?

abo

ut

6.0

yr

8.B

IOLO

GY

In a

lab

orat

ory,

an o

rgan

ism

gro

ws

from

100

to

250

in 8

hou

rs.W

hat

is

the

hou

rly

grow

th r

ate

in t

he

grow

th f

orm

ula

y�

a(1

r)

t ?ab

ou

t 12

.13%

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1.IN

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TIN

GT

he f

orm

ula

A�

P�1

�2t

give

s th

e va

lue

of a

n in

vest

men

t af

ter

tye

ars

in

an a

ccou

nt

that

ear

ns

an a

nn

ual

in

tere

st r

ate

rco

mpo

un

ded

twic

e a

year

.Su

ppos

e $5

00is

in

vest

ed a

t 6%

an

nu

al i

nte

rest

com

pou

nde

d tw

ice

a ye

ar.I

n h

ow m

any

year

s w

ill

the

inve

stm

ent

be w

orth

$10

00?

abo

ut

11.7

yr

2.B

AC

TER

IAH

ow m

any

hou

rs w

ill

it t

ake

a cu

ltu

re o

f ba

cter

ia t

o in

crea

se f

rom

20

to20

00 i

f th

e gr

owth

rat

e pe

r h

our

is 8

5%?

abo

ut

7.5

h

3.R

AD

IOA

CTI

VE

DEC

AY

A r

adio

acti

ve s

ubs

tan

ce h

as a

hal

f-li

fe o

f 32

yea

rs.F

ind

the

con

stan

t k

in t

he

deca

y fo

rmu

la f

or t

he

subs

tan

ce.

abo

ut

0.02

166

4.D

EPR

ECIA

TIO

NA

pie

ce o

f m

ach

iner

y va

lued

at

$250

,000

dep

reci

ates

at

a fi

xed

rate

of

12%

per

yea

r.A

fter

how

man

y ye

ars

wil

l th

e va

lue

hav

e de

prec

iate

d to

$10

0,00

0?ab

ou

t 7.

2 yr

5.IN

FLA

TIO

NFo

r D

ave

to b

uy a

new

car

com

para

bly

equi

pped

to

the

one

he b

ough

t 8

year

sag

o w

ould

cos

t $1

2,50

0.S

ince

Dav

e bo

ugh

t th

e ca

r,th

e in

flat

ion

rat

e fo

r ca

rs l

ike

his

has

been

at

an a

vera

ge a

nn

ual

rat

e of

5.1

%.I

f D

ave

orig

inal

ly p

aid

$840

0 fo

r th

e ca

r,h

owlo

ng

ago

did

he

buy

it?

abo

ut

8 yr

6.R

AD

IOA

CTI

VE

DEC

AY

Cob

alt,

an e

lem

ent

use

d to

mak

e al

loys

,has

sev

eral

iso

tope

s.O

ne

of t

hes

e,co

balt

-60,

is r

adio

acti

ve a

nd

has

a h

alf-

life

of

5.7

year

s.C

obal

t-60

is

use

d to

trac

e th

e pa

th o

f n

onra

dioa

ctiv

e su

bsta

nce

s in

a s

yste

m.W

hat

is

the

valu

e of

kfo

rC

obal

t-60

?ab

ou

t 0.

1216

7.W

HA

LES

Mod

ern

wh

ales

app

eare

d 5�

10 m

illi

on y

ears

ago

.Th

e ve

rteb

rae

of a

wh

ale

disc

over

ed b

y pa

leon

tolo

gist

s co

nta

in r

ough

ly 0

.25%

as

mu

ch c

arbo

n-1

4 as

th

ey w

ould

hav

e co

nta

ined

wh

en t

he

wh

ale

was

ali

ve.H

ow l

ong

ago

did

the

wh

ale

die?

Use

k

�0.

0001

2.ab

ou

t 50

,000

yr

8.PO

PULA

TIO

NT

he

popu

lati

on o

f ra

bbit

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an

are

a is

mod

eled

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grow

th e

quat

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P(t

) �

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wh

ere

Pis

in

th

ousa

nds

an

d t

is i

n y

ears

.How

lon

g w

ill

it t

ake

for

the

popu

lati

on t

o re

ach

25,

000?

abo

ut

4.4

yr

9.D

EPR

ECIA

TIO

NA

com

pute

r sy

stem

dep

reci

ates

at

an a

vera

ge r

ate

of 4

% p

er m

onth

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the

valu

e of

th

e co

mpu

ter

syst

em w

as o

rigi

nal

ly $

12,0

00,i

n h

ow m

any

mon

ths

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tw

orth

$73

50?

abo

ut

12 m

o

10.B

IOLO

GY

In a

lab

orat

ory,

a cu

ltu

re i

ncr

ease

s fr

om 3

0 to

195

org

anis

ms

in 5

hou

rs.

Wh

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s th

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ourl

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owth

rat

e in

th

e gr

owth

for

mu

la y

�a

(1

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abo

ut

45.4

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r � 2

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ctic

e (

Ave

rag

e)

Exp

on

enti

al G

row

th a

nd

Dec

ay

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

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ER

IOD

____

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10-6

10-6


A


Readin

g t

o L

earn

Math

em

ati

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an

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ecay

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____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

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10-6

10-6

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Lesson 10-6

Pre-

Act

ivit

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ow c

an y

ou d

eter

min

e th

e cu

rren

t va

lue

of y

our

car?

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d th

e in

trod

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on t

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esso

n 10

-6 a

t th

e to

p of

pag

e 56

0 in

you

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xtbo

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wh

ich

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o ye

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show

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n t

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tabl

e di

d th

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escr

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way

s to

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cula

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valu

e of

th

e ca

r 6

year

s af

ter

it w

aspu

rch

ased

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o n

ot a

ctu

ally

cal

cula

te t

he

valu

e.)

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ple

an

swer

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Mu

ltip

ly $

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.66

by 0

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su

btr

act

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ltip

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b.

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l co

mpa

red

the

valu

e of

his

car

wh

en h

e bo

ugh

t it

new

to

the

valu

e w

hen

he

trad

ed ‘;

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in

six

yea

rs l

ater

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ecay

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pal

eon

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com

pare

d th

e am

oun

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car

bon

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in t

he

skel

eton

of

an a

nim

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hen

it

died

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amou

nt

300

year

s la

ter.

dec

ay

d.

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epos

ited

$75

0 in

a s

avin

gs a

ccou

nt

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ng

4.5%

an

nu

al i

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pou

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Sh

e di

d n

ot m

ake

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fu

rth

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epos

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Sh

e co

mpa

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bala

nce

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her

pas

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k im

med

iate

ly a

fter

sh

e op

ened

th

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cou

nt

to t

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nce

3

year

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ter.

gro

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2.S

tate

wh

eth

er e

ach

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atio

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epre

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xpon

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row

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cay.

a.y

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Rem

emb

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isu

aliz

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r gr

aph

s is

oft

en a

goo

d w

ay t

o re

mem

ber

the

diff

eren

ce b

etw

een

mat

hem

atic

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quat

ions

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r kn

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of t

he g

raph

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exp

onen

tial

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atio

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om L

esso

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hel

p yo

u t

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mem

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atio

ns

of t

he

form

y�

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r)

t

repr

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pon

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amp

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ph

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bas

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reat

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nct

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incr

easi

ng

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row

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wh

ile if

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1 �

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ase

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less

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Eff

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nn

ual

Yie

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hen

in

tere

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un

ded

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an o

nce

per

yea

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fect

ive

ann

ual

yiel

d is

hig

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th

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ann

ual

in

tere

st r

ate.

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e ef

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ive

ann

ual

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the

inte

rest

rat

e th

at w

ould

giv

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e sa

me

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nt

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nte

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if

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inte

rest

wer

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mpo

un

ded

once

per

yea

r.If

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llar

s ar

e in

vest

ed f

or o

ne

year

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lue

of t

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inve

stm

ent

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end

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year

is

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).If

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or o

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at

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ded

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mes

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valu

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vest

men

t at

th

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d of

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th

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ann

ual

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con

tin

uou

s,th

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lue

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inve

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ent

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he

end

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ne

year

is

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Per

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in s

et t

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al a

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r E

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ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

__P

ER

IOD

____

_

10-6

10-6

Fin

d t

he

effe

ctiv

ean

nu

al y

ield

of

an i

nve

stm

ent

mad

e at

7.5%

com

pou

nd

ed m

onth

ly.

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5

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ctiv

ean

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al y

ield

of

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nve

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ent

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e at

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% c

omp

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sly.

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25

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ple1

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ple1

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ple2

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ple2

Fin

d t

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effe

ctiv

e an

nu

al y

ield

for

eac

h i

nve

stm

ent.

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% c

ompo

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ded

quar

terl

y10

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5% c

ompo

un

ded

mon

thly

8.84

%

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25%

com

pou

nde

d co

nti

nu

ousl

y9.

69%

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75%

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pou

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nti

nu

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y8.

06%

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5% c

ompo

un

ded

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y (a

ssu

me

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5-da

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ar)

6.72

%

6.W

hic

h i

nve

stm

ent

yiel

ds m

ore

inte

rest

—9%

com

pou

nde

d co

nti

nu

ousl

y or

9.

2% c

ompo

un

ded

quar

terl

y?9.

2% q

uar

terl

y

chapter 10 resource masters - north hunterdon-voorhees ... 11/chapter 11... ·...

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