MATH 30-1Exponents & Logarithms Module Six Assignment

Module / Unit 6 - Assignment Booklet

Student:

__________________________________________________

Date Submitted: ___________________________________________

http://moodle.blackgold.ca

Math 30-1: Module 6 Assignment 2 | P a g e

Math 30-1: Module 6 Assignment 3 | P a g e

Lesson 1: Exponential Functions

1. Sketch the graph of and state the domain, range, horizontal asymptote, and intercepts.

2. Complete these tasks for each of the following functions.

i. Describe the transformations.

ii. Sketch the graph of the function.

iii. Identify the domain, range, equation of the horizontal asymptote, and intercepts.

a.

Math 30-1: Module 6 Assignment 4 | P a g e

b.

c.

Math 30-1: Module 6 Assignment 5 | P a g e

3. The radioactive isotope Molybdenum-99 is produced at Chalk River Laboratories. Molybdenum-99 has a half-life of 66 h.

a. What is the exponential function that would describe the relationship between the percent, P, of Molybdenum-99 remaining after t hours?

b. What percent of Molybdenum-99 would be left after 24 h?

Math 30-1: Module 6 Assignment 6 | P a g e

LESSON 1 SUMMARY

In this lesson you explored exponential functions and how they can be used to describe growth or decay of quantities.

The characteristics of all exponential functions of the form y = cx, c > 0, c ≠ 1 are as follows:

o If c > 1, the function is increasing and is an exponential growth function.o If 0 < c < 1, the function is decreasing and is an exponential decay function.o The domain is {x|x ∈ R}.o The range is {y|y > 0, y ∈ R}.o There is no x-intercept.o The y-intercept is 1.o The horizontal asymptote is at y = 0.

When an exponential function is in the form y = a(c)b(x–h) + k, the parameters a, b, h and k correspond to the following transformations:

Value > 0 Value < 0

ao vertical stretch of

the graph by a factor of |a|

o vertical stretch of the graph by a factor of |a|, and a reflection in the x-axis

b

o horizontal stretch of the graph by a

factor of

o horizontal stretch of the graph by a

factor of , and a reflection in the y-axis

ho translated to the

right h unitso translated to the

left |h| units

ko translated up k

unitso translated down |

k| units

Exponential functions can be used to model real-life applications of exponential growth or decay. In the next lesson you will explore how to solve exponential equations

Math 30-1: Module 6 Assignment 7 | P a g e

Lesson 2: Exponential Equations

1. Write the expressions 36x and 2162x 3 so that both expressions have the same base.

2. Solve the exponential equation 32x 83x4 by changing the bases. Check your answer by graphing, and include a sketch of your graph.

Math 30-1: Module 6 Assignment 8 | P a g e

3. Solve the exponential equation 25x4 6252x by changing the bases. Check your answer using substitution.

4. Create your own exponential equation that contains different bases. Include a full solution to your equation.

5. Trinh has decided to save money to buy a new computer. She invests $800 in an account that earns 5.0% per year, compounded quarterly. Determine how long she needs to invest her money until she has $1000. Explain the process you used.

Math 30-1: Module 6 Assignment 9 | P a g e

LESSON 2 SUMMARY

In this lesson you explored how to solve exponential equations.

You learned that when there is a single power on each side of the equal sign and the bases are the same, the exponents are equal. You needed to change the bases of the powers to a common base for some questions.

Another method was to guess and check. This method was useful when the bases could not be changed to a common base.

You can use substitution or graphing to check or verify your solution.

You explored how the compound interest formula could be used to determine how long you would need to invest money in order to have the money grow to a certain amount.

In future lessons you will learn an algebraic method for solving exponential equations when the bases do not share a common base.

Math 30-1: Module 6 Assignment 10 | P a g e

Lesson 3: Understanding Logarithms

1. Rewrite the logarithmic equation in exponential form with positive exponents.

2. Evaluate each logarithm and explain the process used.

a.

b.

c.

Math 30-1: Module 6 Assignment 11 | P a g e

3. Determine the value of a in each of the following equations.

a.

b.

4. In the beginning of this lesson the idea that the growth of a tree could be modelled using a logarithmic function was introduced. A possible logarithmic function that could model the

growth of a tree is h 4 t, where h is the height of the tree in metres and t is the time in years. Estimate the approximate height of the tree to the nearest tenth of a metre after 10 yr. Explain the reasoning for your estimation.

Math 30-1: Module 6 Assignment 12 | P a g e

5. The formula for the Richter magnitude, M, of an earthquake is , where A is the

amplitude of the earthquake and is the amplitude of a standard earthquake with magnitude of 0.

In 2011, an earthquake occurred off the Pacific coast of the Tōhoku region of Japan. This earthquake, which measured 9.0 on the Richter scale, is one of the five most powerful earthquakes in the world since modern record-keeping began in 1900.

By comparison, one of the largest earthquakes to hit Alberta was a magnitude 4.7 earthquake. This earthquake occurred in 1984, on the border between British Columbia and Alberta, just north of Waterton Lakes National Park. Determine how many times larger in amplitude the Tōhoku earthquake was compared to the Alberta quake. Answer to the nearest ten thousand.

Math 30-1: Module 6 Assignment 13 | P a g e

LESSON 3 SUMMARY

In this lesson you explored the concept of logarithms. The inverse of an exponential function, y = cx, c > 0, c ≠ 1, is a logarithmic function, y = logc x, c > 0, c ≠ 1. A logarithm of a number is the exponent by which a given base must be raised to produce that number. You can change between exponential and logarithmic forms.

Exponential Form

Logarithmic Form

x = cy y = logc x

The logarithmic function is the reflection of the exponential function in the line y = x. The characteristics of the logarithmic function y = logc x, c > 0, c ≠ 1 are as follows:

o The domain is {x|x > 0, x ∈ R}.o The range is {y|y ∈ R}.o The x-intercept is 1.o The vertical asymptote is x = 0, or the y-axis.

In the next lesson you will explore the graphs of different logarithmic functions

Math 30-1: Module 6 Assignment 14 | P a g e

Lesson 4: Graphing Logarithmic Functions

1. Consider the functions f(x) and g(x) (x 3) 4.

a. Find the domain and range for both functions.

b. Explain how the graph g(x) can be described as a transformation of f(x).

2. The graph of f(x) x is

vertically stretched about the x-axis by a factor of 2 horizontally stretched about the y-axis by a factor of 3 translated 5 units to the right and 2 units down

If the equation of the transformed image is written in the form determine the values of a, b, c, and d. Then write the equation of g(x).

Math 30-1: Module 6 Assignment 15 | P a g e

3. The slope of a beach is influenced by the local tides, wind and wave patterns, and the size of the sand particles on the beach. In general, as the diameter in millimetres, d, of the average size of the particles on the beach increases, the slope of the beach, s, increases according to the formula s(d) 0.159 0.1181 log d.

Creatas/Thinkstock

a. Describe the graph of the function s(d) as compared to the graph of f (d) log d using transformations terminology.

b. If the slope of a particular beach is 0.005 (about a 1-m vertical climb for every 200 m in the horizontal direction), determine the average diameter (in millimetres) of a grain of sand on the beach. Round your answer to 4 decimal places.

Math 30-1: Module 6 Assignment 16 | P a g e

LESSON 4 SUMMARY

In this lesson you looked at graphing logarithmic functions and how transformations can be applied to logarithmic functions.

When a logarithmic function is in the form the parameters a, b, h, and k correspond to the following transformations of the graph of y = logc x:

Parameter Value > 0 Value < 0

a o vertical stretch by a factor of |a|

o vertical stretch by a factor of |a| and reflection in the x-axis

b o horizontal stretch of

graph by a factor of

o horizontal stretch by a

factor of and reflection in the y-axis

h o translation of h units to the right

o translation of |h| units to the left

k o translation of k units up o translation of |k| units down

In the next lesson you will continue to work with logarithms. You will explore how to work with multiple logarithms in an expression.

Math 30-1: Module 6 Assignment 17 | P a g e

Lesson 5: Laws of Logarithms1. Use the laws of logarithms to evaluate each expression.

a.

b. 108 – 4

c.

d.

Math 30-1: Module 6 Assignment 18 | P a g e

2. Expand each of the following expressions:

a. (2x)

b. (x3)

c.

3. Write each expression as a single logarithm in simplest form. State any restrictions on the variable.

a. (x2 16) (2x 8)

Math 30-1: Module 6 Assignment 19 | P a g e

b. (x2 16) (x2 2x – 8)

c. 3 (x 3) (x2 x 6)

Math 30-1: Module 6 Assignment 20 | P a g e

LESSON 5 SUMMARY

In this lesson you explored the laws of logarithms. You used numeric values to determine what the laws should be and, then, looked at how the laws could be derived. You studied the following laws:

o product law of logarithms:

logb (M × N) = logb M + logb N

o quotient law of logarithms:

o power law of logarithms:

logb (Mn) = nlogb M

The laws of logarithms can be used to evaluate logarithmic expressions as well as expand or simplify logarithmic expressions.

You also saw how logarithmic scales are used. You solved problems involving the decibel scale and the pH scale.

Using the laws of logarithms to expand or simplify logarithmic expressions will be helpful in the next lesson. In Lesson 6 you will solve logarithmic equations.

Math 30-1: Module 6 Assignment 21 | P a g e

Lesson 6: Logarithmic and Exponential Equations1. Provide worked examples for the last row of the concept chart shown.

1

2. The equation can be used to describe the straight-line graph shown. Determine the values of k and n.

1 Source: Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011. Reproduced with permission.

Math 30-1: Module 6 Assignment 22 | P a g e

Math 30-1: Module 6 Assignment 23 | P a g e

3. How long, to the nearest tenth of a year, will it take a $5000 investment to double if the money is invested at 7% interest compounded quarterly? How long will it take to triple the investment?

4. There is a relationship between mortgage amount, number of payments, amount of each payment, how often each payment is made, and the interest rate. The following formula illustrates the relationship:

, where

PV present value i interest rate per compounding period, as a decimal R payment amount n number of payments

Suppose a bank offers you a 3.4% interest rate, compounded semi-annually, on a mortgage amount of $173 112. You pay $1000/mo, or $6000 semi-annually. How many semi-annual payments would you have to make to pay off the mortgage? How long, in years, would it take to pay off the mortgage?

Math 30-1: Module 6 Assignment 24 | P a g e

LESSON 6 SUMMARY

In this lesson you explored how to solve logarithmic and exponential equations.

To solve a logarithmic equation, apply the laws of logarithms to express the left side and the right side of the equation as a single logarithm. You can then use the property if logc A = logc B, then A = B, where c, A, B > 0 and c ≠ 1.

To solve an exponential equation where the bases are not powers of one another, you can use the property if A = B, then logc A = logc B, where c, A, B > 0 and c ≠ 1.

Check that solutions satisfy the original equation. Remember that the logarithm of a number ≤ 0 is undefined.

There are real-world situations that can be modeled with an exponential or logarithmic equation. Some applications involve finances, such as investments and loans. Other applications can be found in science, such as radioactive decay. Many exponential

equations can be expressed in the form

Math 30-1: Module 6 Assignment 25 | P a g e

MODULE 6 – EXPONENTS & LOGARITHMS SUMMATIVE ASSIGNMENTComplete the following questions from your text book. Show steps completely and clearly, as marks are assigned for mathematical literacy and communication. Always use graph paper, rulers, and pencils as necessary. Attach securely to this booklet before you hand everything in.

Module 6 is now complete. Once you have received your corrected work, review your instructor’s comments and prepare for your module six test.

Text: Pre-Calculus 12 - Chapters 7 and 8

Chapter 7 – Exponential Functions

Section 7.1: Page 342 to 345 #7, 10, 11

Section 7.2: Page 354 to 357 #6, 7, 11, 12

Section 7.3: Page 364 to 365 #2, 3, 4, 14

Chapter 8 – Logarithmic Functions

Section 8.1: Page 380 to 382 #2all, 3all, 4all, 12all, 13all, 14a, 15, 20

Section 8.2: Page 389 to 391 #4all, 8, 16

Section 8.3: Page 400 to 403 #1a,c, 2a,c,d, 3a,d, 7a,b,e, 8a,c, 10a, 11a,c,

Section 8.4: Page 412 to 415 #1a,c, 2a,d, 4a,b,5a,c, 7a,d, 8a,d,e 13