unit 3b modeling with logarithmic...

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Name:____________________________

Period: _____

Unit 3B

Modeling with Logarithmic

Functions

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Equivalent Forms of Exponential Expressions Before we begin today’s lesson, how much do you remember about exponents? Use expanded form to write the rules for the exponents.

OBJECTIVE 1 Multiplying Exponential Expressions

32 ∙ 34 𝑦4 ∙ 𝑦10 123 ∙ 125

SUMMARY: 𝑎𝑚 ∙ 𝑎𝑛 = ___________

OBJECTIVE 2 Dividing Exponential Expressions (Remember: 𝑥

𝑥= 1)

36

32

𝑦10

𝑦4

125

123

SUMMARY: 𝑎𝑚

𝑎𝑛 = ___________

OBJECTIVE 3 Negative Exponential Expressions: Simplify 2 WAYS using expanded form AND the rule from OBJECTIVE 2

32

36

𝑦4

𝑦10

123

125

SUMMARY: 1

𝑎𝑛 = ___________

OBJECTIVE 4 Exponential Expressions Raised to a Power

(36)2 (𝑦3)4 (12𝑚)5

SUMMARY: (𝑎𝑚)𝑛 = ___________ SUMMARY: (𝑎 ∙ 𝑏)𝑛 = ____________

OBJECTIVE 5 Same base then the exponents are equal

(2𝑥)2 = 26 444𝑦 = 47 93 = 27𝑥

SUMMARY: 𝑎𝑛 = 𝑎𝑚 then 𝑚 = 𝑛

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Use the rules of exponents to find the value of x in each equation.

1. (212)3 = 22𝑥 2. (11𝑥)(1115) = 11105 3. 4𝑥

434 = 420

4. (510)𝑥 = 550 5. (623)(6𝑥) = 657 6. 322

3𝑥 = 37

7. (62)(36𝑥) = 612 8. (41

2) (161

4) = 𝑥 9. (81

3) (643

2) = 𝑥

10. (253)(59) = 125𝑥 11. (8𝑥)(25) = 47 12. (31

2) (271

2) = 9𝑥

13. 10𝑥 = 10,000 14. 10𝑥+3 = 1,000 15. 103𝑥+2 = 1,000

16. 102𝑥 = 50 17. 3(10)𝑥+4 = 3,000 18. 12(10)3𝑥+2 = 120

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History of Logarithms (It’s a great story)

Quick Multiplication of large numbers

Logarithmic tables

Example: 2.93 x 1.26 From log table log 2.93 = .4669 use log table in reverse (antilog table or log-1) + log 1.26 = .1004 log-1 .5673 = 3.69 .5673 Your Turn: 2.13 x 2.81

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Graphing Remember the chess board from Math 1?

The following story has been told in different ways. Versions of

the story date back over 1000 years.

When the creator of the game of chess showed his invention to

the ruler of the country, the ruler was so impressed that he gave

the inventor the right to choose his own reward. The man asked

the king for the following: that for the first square of the chess

board, he would receive one grain of rice, two grains for the

second square, four on the third square, and so forth, doubling

the amount each time. The ruler quickly accepted the inventor's offer, and thought that the man hadn’t

asked for much. However, the king’s treasurer explained that it would take more than all the rice in the

kingdom to give the inventor the reward. The story usually ends with the inventor becoming the new king

or being executed.

Fill in the amount of rice in each square for the 1st 40 squares

1 21

2 22

3 23

4 24

5 25

6 26

7 27

8 28

9 29

10 30

11 31

12 32

13 33

14 34

15 35

16 36

17 37

18 38

19 39

20 40

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Graph as many grains of rice on the Cartesian plane (Vertical scale by 10’s)

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Graph on semi-log paper

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The Richter Scale The magnitude of an earthquake is a measure of the amount of energy released at its source. The Richter scale is an exponential measure of earthquake magnitude, as shown on the right. The magnitude increases per unit as the energy released increases by powers of ten. A simpler way to examine the Richter Scale is shown below. An earthquake of magnitude 5 releases about 30 times as much energy as an earthquake of magnitude 4. In 1995, an earthquake in Mexico registered 8.0 on the Richter scale. In 2001, an earthquake of magnitude 6.8 shook Washington state. Let’s compare the amounts of energy released in the two earthquakes. For the earthquake in Mexico at 8.0 on the Richter Scale, the energy released is E308 and for the earthquake in Washington state, the energy released is E306.8. A ratio of the two quakes and using the properties of exponents yields the following:

𝑀𝑒𝑥𝑖𝑐𝑜 𝐸𝑎𝑟𝑡ℎ𝑞𝑢𝑎𝑘𝑒

𝑊𝑎𝑠ℎ𝑖𝑛𝑔𝑡𝑜𝑛 𝐸𝑎𝑟𝑡ℎ𝑞𝑢𝑎𝑙𝑒=

𝐸 ∙ 308

𝐸 ∙ 306.8=

308

306.8= 308−6.8 = 301.2 = 59.2

What this means is that the earthquake in Mexico released about 59 times as much energy as the earthquake in Washington. The exponents used by the Richter scale shown in the above example are called logarithms or logs.

Energy Released: X 30

0 1 2 3 6 5 4 9 8 7

Magnitude: +1

E E301 E302 E303 E307 E306 E305 E304 E308 E309

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Introduction to Exponential and Logarithmic Functions Notes

Exponential Function: A function of the form 𝑦 = 𝑎 ∙ 𝑏𝑥, where 𝑎 ≠ 0, 𝑏 > 0, and 𝑏 ≠ 1.

**Exponential Functions are functions whose equations contain a variable in the exponent!!

Exponential Functions have the following characteristics:

The functions is continuous and one-to-one The domain is the set of all real numbers The x-axis is an asymptote of the graph. The range is the set of all positive numbers if 𝑎 > 0 and all negative numbers if 𝑎 < 0. The graph contains the point (0, 𝑎). That is the y-intercept is 𝑎.

The graphs of 𝑦 = 𝑎𝑏𝑥 and 𝑦 = 𝑎 (1

𝑏)

𝑥

are reflections across the y-axis.

Examples: NOT Examples:

Logarithmic Function: The function 𝑥 = 𝑙𝑜𝑔𝑏 𝑦 , where 𝑏 > 0 and 𝑏 ≠ 1, is called a logarithmic

function. This function is the inverse of the exponential function 𝑦 = 𝑏𝑥 and has the following

characteristics:

The function is continuous and one-to-one. The domain is the set of all positive real numbers. The y-axis is an asymptote of the graph. The range is the set of all real numbers. The graph contains the point (1, 0). That is the x-intercept is 1.

Logarithm: In general, the inverse of 𝑦 = 𝑏𝑥 is 𝑥 = 𝑏𝑦 . In 𝑥 = 𝑏𝑦, y is called the logarithm of 𝑥. It is

usually written as 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 and is read “y equals log base b of x.”

The inverse function of the exponential functions with base b, is called the logarithmic function with

base b.

For 𝑥 > 0, 𝑏 > 0, 𝑏 ≠ 0,

EXPONENTIAL FORM LOGARITHM FORM

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Rewriting in both forms.

Example 1) Rewrite logarithmic each equation in its equivalent exponential form.

a. log5 𝑥 = 2 d. 3 = log𝑏 64

b. log3 7 = 𝑦 e. 3 = log7 𝑥

c. 2 = log𝑏 25 f. log4 26 = 𝑦

Example 2) Rewrite each exponential equation in its equivalent logarithmic form.

a. 122 = 𝑥 d. 𝑏3 = 8

b. 25 = 𝑥 e. 𝑏3 = 27

c. 83 = 𝑐 f. 4𝑦 = 9

Basic and Inverse Log Properties- Because logs are exponents, they have properties that can be

verified using the properties of exponents.

Basic Properties: Inverse Properties: (Cancel with the same base!)

log𝑏 𝑏 = 1 because 𝑏1 = 𝑏 log𝑏 𝑏𝑥 = 𝑥

log𝑏 1 = 0 because 𝑏0 = 1 𝑏log𝑏 𝑥 = 𝑥

Example 3) Evaluate using the log properties.

a. 𝑙𝑜𝑔7 7 e. 𝑙𝑜𝑔9 9

b. 𝑙𝑜𝑔5 1 f. 𝑙𝑜𝑔8 1

c. 𝑙𝑜𝑔4 45 g. 6𝑙𝑜𝑔6 9

d. 𝑙𝑜𝑔7 78 h. 3𝑙𝑜𝑔3 17

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Common Logarithm: Base 10 Logarithm, usually written without the subscript 10.

log10 𝑥 = log 𝑥 , 𝑥 > 0. Most calculators have a LOG key for evaluating common logarithms.

The calculator is programmed in base 10.

Example 4) Find the value of each log. Round to the nearest ten-thousandths.

a. 𝑙𝑜𝑔 81,000 c. 𝑙𝑜𝑔 0.35

b. 𝑙𝑜𝑔 6 d. 𝑙𝑜𝑔 0.0027

Evaluating Logs using the Change of Base Formula

For all positive numbers, a, b, and n, where 𝑎 ≠ 1 𝑎𝑛𝑑 𝑏 ≠ 1,

log𝑎 𝑛 = log𝑏 𝑛

log𝑏 𝑎 Example: log5 12 =

log10 12

log10 5

This formula allows us to evaluate a logarithmic expression of any base by translating the

expression into one that involves common logarithms.

Example 5) Evaluate each logarithm

a. log3 18 d. log25 5

b. log4 25 e. log2 1024

c. log2 16 f. log5 125

Solving for variables with exponentials and logs.

****MAY HAVE TO REWRITE AND APPLY PROPERTIES OR CHANGE OF BASE FORMULA!!!

Example 6) Solve for the variable:

a. log3 243 = 𝑦 b. log9 𝑥 = −3 c. log8 𝑛 =4

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Example 7) Evaluate:

a. log8 84 = 𝑥 b. log9 92 = 𝑦

Example 8) Solve each log equation. Be sure to check your answers!

a. log3(3𝑥 − 6) = log3(2𝑥 + 1) b. log8(𝑥2 − 14) = log8(5𝑥)

c. log4 𝑥2 = log4(4𝑥 − 3) d. log5(𝑥 − 7) = 2

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The Meaning of Logarithms (Kuta Software)

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Graphing and Transforming Exponential Functions

Part I

Part 2:

Horizontal Asymptote: an “invisible” line that the graph approaches as x goes to infinity or negative infinity.

Identify the horizontal asymptote of (a): Identify the horizontal asymptote of (b): Identify the y-intercept of (a): Identify the y-intercept of (b): End behavior: End Behavior:

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Transforming Exponential Functions

Translate left or right: Vertical stretch or compression: Horizontal stretch or compression: Reflections: Translate up or down:

Let’s look at the following example. The graph on the right represents a transformation of the graph of f(x) = 3 log10 x + 1.

x = 3: Stretches the graph vertically. h = 0: There is no horizontal shift. k = 1: The graph is translated 1 unit up.

Graph the following function on the graph at right. Describe each transformation, give the domain and range, and identify any asymptotes. y = -2log10(x + 2) – 4 Domain: Range: Asymptote: Description of transformations:

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Independent Practice Graph the following transformations of the function y = log10 x on the coordinate planes. Determine the domain, range, and asymptotes of each transformation. Describe the transformations. 1) y = log10 x – 6 6) y = -log10 (x + 2) 7) y = log10 2x

Domain: Domain: Domain: Range: Range: Range: Asymptotes: Asymptotes: Asymptotes:

Description: Description: Description:

2) Describe in your own words what happens to the graph of f(x) = log2 x under the given

transformations, then graph. f(x) = log2 (x – 2) b. f(x) = log2 (x) + 3 c. f(x) = log2 (x – 2) + 3

3) State the domain, range, intercepts and asymptotes of f(x) = log2 (x – 2) + 3. Domain:

Range:

Intercept(s):

Asymptote(s):

4) Describe the transformations of y = 4 log2 (2x – 4) + 6 from the parent function y = log2 (x).

5) Describe the transformations of y = -3 log 10 (4x + 3) – 2 from the parent function

y = log10 (x).

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Investigation 2: Solving for Exponents Logarithms can be used to find exponents that solve equations like 10x = 9.5. For this reason, they are an invaluable tool in answering questions about exponential growth and decay. For example, the world population is currently about 6.2 billion and growing exponentially at a rate of about 1.14% per year. To find the time when this population is likely to double, you need to solve the equation 6.2(1.0114)t = 12.4, or (1.0114)t = 2 As you work on the problems of this investigation, look for ways to answer this question:

How can common logarithms help in finding solutions of exponential equations?

1) Use number sense and what you already know about logarithms to solve these equations. a. 10x = 1,000 b. 10x + 2 = 1,000

c. 103x + 2 = 1,000 d. 2(10)x = 200

e. 3(10)x + 4 = 3,000 f. 102x = 50

g. 103x + 2 = 43 h. 12(10)3x + 2 = 120

i. 3(10)x + 4 + 7 = 28

Use the rules of exponents or logarithms to find the value of x in each equation.

1. (5𝑥+1)5 = 525𝑥 2. (92𝑥)(916) = 948 3. 450

440 = 4𝑥−5

4. (642)(16𝑥) = 412 5. (161

2) (23) = 𝑥 6. (253)(56) = 1253𝑥−2

7. 102𝑥−1 = 100 8. -2(10)𝑥+4 = −.002 9. (10)𝑥 = 1

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Use the rules of exponents or logarithms to find the value of x in each equation.

1. (212)3 = 22𝑥 2. (11𝑥)(1115) = 11105 3. 4𝑥

434 = 420

4. (510)𝑥 = 550 5. (623)(6𝑥) = 657 6. 322

3𝑥 = 37

7. (62)(36𝑥) = 612 8. (41

2) (161

4) = 𝑥 9. (81

3) (643

2) = 𝑥

10. (253)(59) = 125𝑥 11. (8𝑥)(25) = 47 12. (31

2) (271

2) = 9𝑥

13. 10𝑥 = 10,000 14. 10𝑥+3 = 1,000 15. 103𝑥+2 = 1,000

16. 102𝑥 = 50 17. 3(10)𝑥+4 = 3,000 18. 12(10)3𝑥+2 = 120

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Investigating the Properties of Logarithms **For the purpose of this activity we will be using Common Logs.

PART I Complete the following table using your calculator. Round answers to four decimal places.

log 5 0.6990

log 10 – log 2

log 3

log 18 – log 6

log 7

log 28 –log 4

log ½

Log 3 – log 6

log 2

Log 8 – log 4

1. Using any patterns you see in the results above, what generalizations could be made? 2. How could we find the value of log 3 if the “3” button is missing from our calculator? PART II Complete the following table using your calculator. Round answers to four decimal places.

log 12

log 6 + log 2

log 18

log 3 + log 6

log 9

log 3 + log 3

log 20

log 4 + log 5

log 26

log 2 + log 13

1. Using any patterns you see in the results above, what generalizations could be made? 2. Have you noticed a similar result before? Where? 3. How could we find the value of log 30 if the “3” button is missing from our calculator?

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PART III Consider the following table.

Number Equivalent value with a different base

16 24, 42

64 26, 43, 82

81 24 9,3

49 27

25 25

Complete the following table using your calculator. Round values to four decimal places.

4log 2 1.2041

log 16

2log 5

log 25

3log 4

6log 2

log 64

2log 7

log 49

1. Using the above tables, what generalization(s) can be made? Can these generalizations be linked to your previous knowledge of exponents? How? 2. A student noticed that log ½ gave the same value as –log 2. How is this possible?

3. Summarize all the properties of logarithms you know. Compare your results with others in the class.

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Investigation : Exponential Growth & Decay Materials Needed:

Graphing Calculator (to serve as a random number generator) To use the calculator’s random integer generator feature:

1. Type any number besides zero into your calculator,

press STO , MATH, , ENTER, ENTER

2. Press MATH 5 1 , 6 )

You can use numbers other than 1 and 6 here. The calculator will choose numbers between and including these numbers when you press enter. Continue pressing enter for more numbers.

Investigation: 1. Choose a recorder to collect the class’s data on the board. You’ll copy the data down in your table later. 2. Everyone in the class should stand so that the recorder can count everyone and record the number of people standing in

a table for “Stage 0”. 3. Use your calculator to find a random integer between 1 and 6. If you roll a 1, sit down. Before proceeding, allow time for

the recorder to count the number of people still standing. When the recorder is finished counting, (s)he will let you know.

4. Repeat step 3 until fewer than 3 people are standing (or you run out of room on the table). 5. Record the data in your table.

Stage 0 1 2 3 4 5 6 7 8 9 10

People Standing

Questions: 1. What is your initial value for this set of data? What does it represent in the investigation? 2. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 3. Based on your answer to Question 2, find the r OR d for the data you collected. Show the process you used to do so. 4. Could you estimate your answer to Question 3 without conducting the exploration? If so, how? 5. Write a recursive (NOW-NEXT) function that would help you make predictions for this data.

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6. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the stage of the investigation and let f(x) equal the number of people standing in that stage.

Investigation 2: Half of a radioactive substance decays every 53 years. How much will remain of a 12 milligram sample after 530 years?

Complete the table.

Years 0 53 106 159 212 265 318 371 424 477 530

Remaining radioactive substance

Questions: 7. What is your initial value for this set of data? What does it represent in the investigation? 8. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 9. Based on your answer to Question 8, find the r OR d for the data you collected. Show the process you used to do so. 10. Could you estimate your answer to Question 9 without filling in the table? If so, how? 11. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 12. Write an explicit function using function notation that would help you make predictions for this data. In your function let

x be the number of 53 year increments in the investigation and let f(x) equal the amount of radioactive substance remaining.

13. Write an explicit function using function notation that would help you make predictions for this data. In your function let

x be the number of years in the investigation and let f(x) equal the amount of radioactive substance remaining. Use your equation to determine how much radioactive substance will remain after 500 years.

14. When will there be exactly 5 milligrams of the radioactive substance? Determine your answer to the nearest month. 15. Compare Investigation 1 and Investigation 2. What are the similarities and differences?

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Investigation 3: You invest money in a savings account that earns 2.5% interest annually. How much money will you have at the end of 10 years if you begin with $1000?

Complete the table.

Years 0 1 2 3 4 5 6 7 8 9 10

Money in your account

Questions: 16. What is your initial value for this set of data? What does it represent in the investigation? 17. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 18. Based on your answer to Question 17, find the r OR d for the data you collected. Show the process you used to do so. 19. Could you estimate your answer to Question 18 without filling in the table? If so, how? 20. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 21. Write an explicit function using function notation that would help you make predictions for this data. In your function let

x be the number of years in the investigation and let f(x) equal the amount of money in the account. 22. You find a bank that will pay you 3% interest annually, so you consider moving your account. Your current bank decides

you’re a good customer and offers you a special opportunity to compound your interest semiannually‼! (They make it sound like it’s a really good deal, so you’re curious). You don’t play around with your money, so you ask what exactly that means. They explain that you’ll still get 2.5% interest, but they’ll give you 1.25% interest at the end of June and 1% interest at the end of December. You want to see if you make more money than you would if you just switched banks, so you do the calculations. Which bank is giving you a better deal? Explain your answer.

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When you write an equation for a situation and use it to make predictions, you assume that other people who use it will understand the situation as well as you do. That’s not always the case when you take away the context, so we sometimes need to provide some additional information to accompany the equation.

23. The domain of a function is the set of all the possible input values that can be used when evaluating it. If you remove

your functions in Questions 6 and 13 and 21 from the context of this situation and simply look at the table and/or graph

of the function, what numbers are part of the theoretical domain of the function?

Will this be the case with all exponential functions? Why or why not?

24. When you consider the context, however, not all of the numbers in the theoretical domain really make sense. We call the

numbers in the theoretical domain that make sense in our situation the practical domain. For instance, in the first

investigation, our input values are “Stages”. If you look at the tables you created, what numbers would be a part of the practical domain for these investigations?

Investigation 1 Investigation 2 Investigation 3 Make note of any similarities and differences and explain why they exist.

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Question Exponential Growth or

Decay?

Write a function that represents this situation

Answer:

1. You buy a house for $130,000. It appreciates 6% per year. How much is it worth in 10 years?

Initial Amount =

Growth/Decay Rate: Percent = Decimal =

Function

2. Justin Beiber is losing 20% of his hair each year. If he currently has 1,546 hairs on his head, about how many hairs will he have left after 10 years?

Initial Amount =


Function

3. If you invest $40 in an account for 10 years at a 3% interest rate compounded semi-annually, how much money will you have?

Initial Amount =


Function

4. A population of 100 frogs increases at an annual rate of 22%. How many frogs will there be in 5 years?

Initial Amount =


Function

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Question Exponential Growth or

Decay?

Write a function that represents this situation

Answer:

5. A species of extremely rare, deep water fish are slowly becoming extinct. If there are a total 821 of this type of fish and there are 15% fewer fish each month, how many will there be in half a year?

Initial Amount =


Function that represents this situation:

6. The population of Austin is growing at a rate of 5% per year. In 2010, the population was 500,000. What would be the predicted current population?

Initial Amount =



7. Use the equation from the previous question and predict in what year Austin’s population will first reach 1,000,000.

8. Carbon-14 has a half-life of 5,730 years. If a fossil that originally had 500 mg of carbon-14 is found and determined to be 27,000 years old, how much carbon-14 was left?

Initial Amount =



9. A super-deadly strain of bacteria is causing the zombie population to double every 2 days. Currently, there are 25 zombies. After how many days will there be 25,600 zombies? (Careful with this one!!!)

Initial Amount =



unit 3b modeling with logarithmic...

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