Chapter 3 – Analyzing Exponential and LogarithmicFunctions Answer Key3.1 Functions and Inverses

Answers1. Not inverses2. Not inverses3. Inverses4. Inverses5. Inverses

6.

7.

8. 9.

10.

11.

12.

13. Inverse:

14. Inverse: 15. The error is in step 2, both ’s should have been replaced with ’s

3.2 One-to-One Functions and Their Inverses

Answers1. Horizontal Line test - if any horizontal line intersects the graph of the function more

than once, then the function is not one to one. If no horizontal line intersects the graph of

than once, then the function is not one to one. If no horizontal line intersects the graph ofthe function more than once, then the function is one-to-one.

2. A function is said to be one-to-one if and only if stating leads directly to

3. Not a 1-1 function4. Not a 1-1 function5. Is a 1-1 function6. 12, 6, 187. 20, 21, 1108. is not 1-to-19. is 1-to-110. is 1-to-111. , , is not 1-to-112. not a 1-1 function13. This graph represents a function, but (-4, 2) and (2, 2) do not pass horizontal line test

so it is not a 1 to 1 function14. 1 to 1, and it is a function15. This is a function but it is not a 1 to 1 function

3.3 Basic Exponential Functions

Answers1. An exponential function is a function with the variable (x) in the exponent:2. An exponential function with increasing outputs is an exponential growth function.3. An exponential function with decreasing outputs is an exponential decay function.4. 125. 320

6.

7. 8.

9.

10.

11. Average rate of change = 3/56 (at , and at , )

12. $123,194.11 13. $22,351,421,771.88 14. $16,465.36

15. = 1.1 grams

16. 4.86 grams 17.

3.4 Graphs of Exponential Functions

3.4 Graphs of Exponential Functions

Answers1.

2.

3.

4.

5.

6.

7. will be moved to the right 4 spaces.

8. will be reflected over the x axis.

9. will be moved down 2 spaces.

10. will move left 2 spaces and be reflected over the x axis.

11. will be moved down 3 and right 4.

12. Stretch horizontally by a factor of 3

13. Invert and stretch vertically by a factor of 3

14. Compress vertically by 5

15. Invert, compress vert 5, and horiz 3, shift down 4

3.5 Logarithmic Functions

Answers1. The logarithm of two hundred fourty-three with base three.

2. 3. Yes, they are equivalent. The first is in log form and the second is in exponential form4.

4. 5. 6. 7. because .8. because

9. because

10. because 11. Bases are the same, ignore the log and solve for : 12. 13. 14. 15.

3.6 Graphs of Logarithmic Functions

AnswersIdentify the domain and range, then sketch the graph.

1. Domain: x>1 Range: All Real Numbers

2. Domain x>1 Range: All Real Numbers

3. Domain: x > -11/3 Range: All real numbers

4. Domain: x> -14/3 Range: All Real Numbers

5. Domain: x>-1 Range: All Real Numbers

Look at the graph and identify the function that the graph represents from the functions listedbelow. 6. f(x) = log x

7. f(x) = -log x

7. f(x) = -log x

8. f(x) = log x+3

9. f(x) = log (-3x)

10. f(x) = -log (-2x)

Graph the following sets of logarithms. Determine if any of them are the same, and if they shouldbe.

11.

12.

13.

14 - 19Point onexponential curve

Corresponding pointon logarithmic curve

(-3, 1/8) (1/8, -3)(-2, 1/4) (1/4, -2)(-1, 1/2) (1/2, -1)

(0, 1) (1, 0)(1, 2) (2, 1)(2, 4) (4, 2)(3, 8) (8, 3)

Graph logarithmic functions, using the inverse of the related exponential function. Then graph thepair of functions on the same axes.

20 – 21

3.7 Properties of Logarithms

Answers1.

2. 3. 4.

5. 6. 7.

8. 9. 10.

11.

12.

13.

14. or 15. 16.

3.8 Common and Natural Logarithms

Answers1. Common logarithms are logarithms with a base of 10. They are most commonly used in the

scientific community.2. A natural logarithm is a logarithm with a base of e. The natural log is used, among other things,

in finance and banking.3. 4.22284. 2.65135. 3.75266 1.60567 28 2.33489 3.445410 -0.154611. 4.025412. 7.600913. 6.856614. 0.105415. 116. or or 17. 18. or -8.155

3.9 Exponential Models

Answers$6,381.41$18,895.68$4,438.85$10,137.65$80,175.12

a) 1.07

b) 61,118

a) 1.03b) 33,319

a) 1.02b) 216,451

a) 1.043 b) 5,889,810

a) 1.01067b) 1,546,659

a) .88b) $2,111

a) .84b) $4,131

a) .79b) $15,775

a) .935b) $7,390

a) .9533b) $183,732

8.16 years14.29 years6.49 years13.51 years12 years

3.10 Logarithmic Models

Answers

1.2.3.4.5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.17.18.19.20.

1.

2.

3.

4.

5.6.

886253643x = 2

no solution

3.11 Simple and Compound Interest

Answers1. Simple interest formula is

2. Compound interest formula is 3. They would have earned: $360.004. Kyle’s balance would be: $1080.005. Yearly simple interest is effectively the same as compound interest compounding yearly.

Roberta would have6. The bank has been paying 4% annually7. She could expect to have $3,122.008. The account has been earning interest for 3 years9. There is a balance of $885.0810. Her original deposit was $650.0011. His yearly interest rate is 8%

12. 13. Georgia has $3570.00, Kirk has $4500.00, and Lottie has $3810.0014. , $7500.0015. He will owe a total of: $318.2716. $60017. $715.5618. $15,300.14 (Reasonably complex: requires 5 iterations of A = P r^t and .3 x interest

each iteration subtracted from the balance)

19. $13,138.75 ($ )

3.12 Exponential Decay

Answers

6.7.

8.9.10.11.12.13.14.15.16.

17.18.

19.

Answersyesno

Ai = 10 | r = .2 (20% decay rate)Ai = 18 | r = .11 (89% decay rate)Ai = 2 | r = .25 (75% decay rate)

1.278 gramsCalculate the time using half of the original amount of carbon as Af :

is the appropriate equationapx 5,728.5 years

3.13 Exponential Growth

Answers$

$ = $3684.61

$3714.23$5709.905.5%

$1,502,083Apx 7.4 years9.35 billion

6,080

or nearly 27 quadrillion

$4,880$7,263.39

CK-12 Math Analysis Concepts 27

1.2.3.4.5.6.7.8.9.10.11.12.13.14.

1)2)

15.

1.2.3.4.5.6.7.8.9.10.11.

12. a.b.

13. a.b.