Deck 12: Logarithmic and Exponential Functions

Question 1

Solve for x.
$$5 ^ { x } = \frac { 1 } { 25 }$$

A) $x = \frac { 1 } { 5 }$
B) $x = - 2$
C) $x = \frac { 1 } { 2 }$
D) $x = 2$

Answer 1

B

Question 2

Graph the function.
$$f ( x ) = e ^ { 6 x }$$

A)
B)
C)
D)

Answer 2

A

Question 3

Graph the function.
$(x)=4^{x-3}$

A)
B)
C)
D)

Answer 3

C

Question 4

Graph the function.
$$f ( x ) = e ^ { x }$$

A)
B)
C)
D)

Question 5

Graph the function.
$f(x)=2^{x}$

A)
B)
C)
D)

Question 6

Graph the function.
$$f ( x ) = 4 ^ { - x }$$

A)
B)
C)
D)

Question 7

Solve for x.
How much money will Lyle have in 6 years is he invests $\$ 9000$ at $17 \%$ annual rate of interest compounded quarterly? How much will he have if it is compounded monthly? Use the interest formula $\mathrm { A } = \mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { n } } \right) ^ { \mathrm { n } t }$ . Round the answers to the nearest cent.

A) $\$ 9964.30$ when compounded quarterly
$\$ 9965.73$ when compounded monthly
B) $\$ 24,438.13$ when compounded quarterly
$\$ 67,279.74$ when compounded monthly
C) $\$ 9964.30$ when compounded quarterly
$\$ 24,780.75$ when compounded monthly
D) $\$ 24,438.13$ when compounded quarterly
$\$ 24,780.75$ when compounded monthly

Question 8

Graph the function.
$$f ( x ) = e ^ { x - 1 }$$

A)
B)
C)
D)

Question 9

Solve for x.
$$4 - 3 x + 5 = \frac { 1 } { 256 }$$

A) $x = 128$
B) $x = \frac { 1 } { 64 }$
C) $x = 3$
D) $x = - 3$

Question 10

Solve for x.
$$2 ^ { x } = 16$$

A) $x = 5$
B) $x = 8$
C) $x = 4$
D) $x = 3$

Question 11

Solve for x.
Mary is investing $\$ 8000$ at an annual rate of $5.4 \%$ compounded annually. How much money will Mary have after 6 years? Use the interest formula $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ . Round the answer to the nearest cent. $\begin{array} { l l } \text { A) } \$ 50,592.00 & \text { B) } \$ 10,592.00 \end{array}$
B) $\$ 10,592.00$
C) $\$ 106,712.26$
D) $\$ 10,968.16$

Question 12

Solve for x.
$$3 ^ { x } = 1$$

A) $x = 1$
B) $x = 0$
C) $x = \frac { 1 } { 8 }$
D) $\varnothing$

Question 13

Solve for x.
$$4 x - 1 = 1024$$

A) $x = 4$
B) $x = 2$
C) $x = 1$
D) $x = 3$

Question 14

Graph the function.

A)
B)
C)
D)

Question 15

Solve for x.
$$4 - 2 x + 10 = 256$$

A) $x = 5$
B) $x = 3$
C) $x = 64$
D) $x = - 3$

Question 16

Solve for x.
$$2 ^ { 3 x + 7 } = \frac { 1 } { 4 }$$

A) $x = 1$
B) $x = \frac { 1 } { 2 }$
C) $x = - 3$
D) $x = 3$

Question 17

Solve for x.
$$4 ^ { - x } = \frac { 1 } { 16 }$$

A) $x = 2$
B) $x = \frac { 1 } { 2 }$
C) $x = - 2$
D) $x = \frac { 1 } { 4 }$

Question 18

Graph the function.
$$f ( x ) = 4 ^ { x } + 2$$

A)
B)
C)
D)

Question 19

Solve for x.
$$3 ^ { 2 x + 1 } = 27$$

A) $x = - 1$
B) $x = 9$
C) $x = 3$
D) $x = 1$

Question 20

Solve for x.
The number of bacteria in a culture is given by $B ( t ) = 8000 \left( 2 ^ { t } \right)$ , where $t$ is the time in hours. How many bacteria will grow in the culture in the first 8 hours?

A) $2,048,000$
B) 224,000
C) 128,000
D) 80,000

Question 21

Solve for x.
The radioactive decay of radon 222 can be described by the equation $\mathrm { A } = \mathrm { Ce } \mathrm { e } ^ { - 0.1813 \mathrm { t } }$ , where $\mathrm { C }$ is the original amount of radon and $\mathrm { A }$ is the amount of radon after $\mathrm { t }$ days. If $2.5$ milligrams are in a laboratory container today, how much was there is the container 11 days ago? Round to the nearest hundredth.

A) $2.84 \mathrm { mg }$
B) $0.34 \mathrm { mg }$
C) $45.68 \mathrm { mg }$
D) $18.37 \mathrm { mg }$

Question 22

Solve.
$$\log _ { 4 } \frac { 1 } { 64 } = x$$

A) $x = - 3$
B) $x = \frac { 1 } { 256 }$
$C ) x = \frac { 1 } { 16 }$
D) $x = 3$

Question 23

Write in exponential form.
$$\log _ { 3 } 9 = x$$

A) $x = 2$
B) $x = 27$
C) $x = 12$
D) $x = 3$

Question 24

Write in exponential form.
$$\text { -3 = } \log _ { 10 } 0.001$$

A) $100.001 = - 3$
B) $0.001 ^ { - 3 } = 10$
C) $- 3^{10} = 0.001$
D) $10 ^ { - 3 } = 0.001$

Question 25

Write in logarithmic form.
$$343 = 7 ^ { 3 }$$

A) $\log _ { 7 } 343 = 3$
B) $\log _ { 3 } 343 = 7$
C) $\log _ { 7 } 3 = 343$
D) $\log _ { 343 } 7 = 3$

Question 26

Write in logarithmic form.
$$100,000 = 10 ^ { 5 }$$

A) $\log _ { 5 } 100,000 = 10$
B) $\log _ { 10 } 100,000 = 5$
C) $\log _ { 5 } 10 = 100,000$
D) $\log _ { 10 } 5 = 100,000$

Question 27

Write in logarithmic form.
$$8 = 2 ^ { 3 }$$

A) $\log _ { 8 } 2 = 3$
B) $\log _ { 2 } 3 = 8$
C) $\log _ { 3 } 8 = 2$
D) $\log _ { 2 } 8 = 3$

Question 28

Write in logarithmic form.
$$\frac { 1 } { 196 } = 14 ^ { - 2 }$$

A) $\log _ { 14 } \frac { 1 } { 196 } = - 2$
B) $\log _ { 14 } \frac { 1 } { 196 } = 14$
C) $\log _ { 14 } ( - 2 ) = \frac { 1 } { 196 }$
D) $\log \frac { 1 } { 196 } 14 = - 2$

Question 29

Write in exponential form.
$$- 3 = \log _ { 2 } \frac { 1 } { 8 }$$

A) $2 ^ { 8 } = 3$
B) $2 ^ { - 3 } = \frac { 1 } { 8 }$
C) $\left( \frac { 1 } { 8 } \right) ^ { 3 } = 2$
D) $3 ^ { 2 } = \frac { 1 } { 8 }$

Question 30

Write in logarithmic form.
$$\frac { 1 } { 16 } = 4 ^ { - 2 }$$

A) $\log _ { 4 } \frac { 1 } { 16 } = - 2$
B) $\log _ { - 2 } \frac { 1 } { 16 } = 4$
C) $\log _ { 1 / 16 } 4 = - 2$
D) $\log _ { 4 } - 2 = \frac { 1 } { 16 }$

Question 31

Write in exponential form.
$$3 = \log _ { 5 } x$$

A) $5 ^ { x } = 3$
B) $5 ^ { 3 } = x$
C) $3 ^ { 5 } = x$
D) $x ^ { 3 } = 5$

Question 32

Solve for x.
$$y = 180 ( 2.7 ) ^ { 0.07 t }$$

A) 402
B) 428
C) 415
D) 408

Question 33

Solve for x.
The rabbit population in a forest area grows at the rate of 7% monthly. If there are 180 rabbits in September, find how many rabbits (rounded to the nearest whole number)should be expected by next September. Use The radioactive decay of radium 226 can be described by the equation $\mathrm { A } = \mathrm { Ce } - 0.0004279 \mathrm { t }$ , where $\mathrm { C }$ is the original amount of radium and $\mathrm { A }$ is the amount of radium remaining after $\mathrm { t }$ years. If 22 milligrams of radium are sealed in a container now, how much radium will be in the container after 100 years. Round to the nearest thousand th.

A) $1.191 \mathrm { mg }$
B) $21.991 \mathrm { mg }$
C) $21.078 \mathrm { mg }$
D) $22.962 \mathrm { mg }$

Question 34

Write in exponential form.
$$- 6 = \log _ { 10 } ( 0.000001 )$$

A) $0.000001 ^ { - 6 } = 10$
B) $10 ^ { - 6 } = 0.000001$
C) $0.000001 ^ { 10 } = - 6$
D) $- 6^{10} = 0.000001$

Question 35

Write in exponential form.
$$\frac { 1 } { 2 } = \log _ { 16 } 4$$

A) $16 ^ { 4 } = \frac { 1 } { 2 }$
B) $4 ^ { 1 / 2 } = 16$
C) $16 ^ { 1 / 2 } = 4$
D) $\left( \frac { 1 } { 2 } \right) ^ { 16 } = 4$

Question 36

Solve for x.
A city is growing at the rate of $0.3 \%$ annually. If there were $5,226,000$ residents in the city in 2008 , find how many (to the nearest ten-thousand) are living in that city in 2012 . Use $y = 5,226,000 ( 2.7 ) ^ { 0.003 t }$ .

A) $5,290,000$
B) $5,320,000$
C) $14,110,000$
D) 170,000

Question 37

Write in exponential form.
$$3 = \log _ { 5 } 125$$

A) $5 ^ { 3 } = 125$
B) $3 ^ { 5 } = 125$
C) $125 ^ { 3 } = 5$
D) $5 ^ { 125 } = 3$

Question 38

Write in logarithmic form.
$$\mathrm { y } = \mathrm { e } ^ { - 2 }$$

A) $\log _ { e } - 2 = y$
B) $\log _ { y } e = - 2$
C) $\log _ { e } y = - 2$
D) $\log _ { - 2 } y = e$

Question 39

Write in logarithmic form.
$$0.001 = 10 ^ { - 3 }$$

A) $\log _ { - 3 } 10 = 0.001$
B) $\log _ { 10 } ( - 3 ) = 0.001$
C) $\log _ { 10 } ( 0.001 ) = - 3$
D) $\log _ { - 3 } ( 0.001 ) = 10$

Question 40

Write in exponential form.
$$0 = \log _ { 13 } 1$$

A) $13 ^ { 0 } = 1$
B) $1 ^ { 13 } = 0$
C) $0 ^ { 13 } = 1$
D) does not exist

Question 41

Evaluate.
$\log _ { 10 } ( 0.01 )$

A) $x = - 20$
B) $x = 100$
C) $x = 2$
D) $x = - 2$

Question 42

Solve.
$$\log _ { 5 } x = 3$$

A) $x = 125$
B) $x = 8$
C) $x = 243$
D) $x = 15$

Question 43

Evaluate.
$2 ^ { \log _ { 2 } 13 }$

A) $x = 15$
B) $x = \log _ { 2 } 13$
C) $x = 13$
D) $x = 2$

Question 44

Evaluate.
$\log _ { 7 } 7$

A) $x = \frac { 1 } { 7 }$
B) $x = 1$
C) $x = 0$
D) $x = 7$

Question 45

Evaluate.
$$\log _ { 10 } 100$$

A) $x = - 2$
B) $x = 2$
C) $x = 20$
D) $x = \frac { 1 } { 100 }$

Question 46

Solve.
$\log _ { 25 } 5 = w$

A) $w = 2$
B) $w = \frac { 1 } { 2 }$
C) $w = 25$
D) $w = 5$

Question 47

Evaluate.
$\log _ { 5 } \sqrt [ 6 ] { 5 }$

A) $x = 15,625$
B) $x = \frac { 1 } { 6 }$
C) $x = 6$
D) $x = \frac { 1 } { 15,625 }$

Question 48

Solve.
$$\log _ { 7 } 1 = x$$

A) $x = 0$
B) $x = 7$
C) $x = 1$
D) $x = 49$

Question 49

Solve.
$$\log _ { x } 125 = 3$$

A) $x = - 5$
B) $x = - 3$
C) $x = \frac { 1 } { 125 }$
D) $x = 5$

Question 50

Solve.
$\log _ { a } 1000 = 3$

A) $a = 1000$
B) $a = 10$
C) $a = 3$
D) $a = 10 ^ { 3 }$

Question 51

Evaluate.
$$\log _ { 5 } \sqrt { 5 }$$

A) $x = 5$
B) $x = \frac { 1 } { 2 }$
C) $x = 1$
D) $x = \frac { 1 } { 5 }$

Question 52

Evaluate.
$\log _ { 5 } 125$

A) $x = 3$
B) $x = \frac { 1 } { 3 }$
C) $x = 1$
D) $x = 15$

Question 53

Evaluate.
$\log _ { 3 } \frac { 1 } { 9 }$

A) $x = - 2$
B) $x = 6$
C) $x = 2$
D) $x = \frac { 1 } { 2 }$

Question 54

Graph.
$y=\log _{2} x$

A)

B)
C)
D)

Question 55

Evaluate.
$\log _ { 11 } 1$

A) $x = 1$
B) $x = 11$
C) $x = \frac { 1 } { 11 }$
D) $x = 0$

Question 56

Solve.
$$\log _ { 9 } x = 1$$

A) $x = 9$
B) $x = \frac { 1 } { 9 }$
C) $x = 1$
D) $x = 0$

Question 57

Solve.
$\log _ { 6 } \left( \frac { 1 } { 36 } \right) = y$

A) $y = - 2$
B) $y = 2$
C) $y = - 36$
D) $y = 6$

Question 58

Evaluate.
$\log _ { 64 } 4$

A) $x = 1$
B) $x = \frac { 1 } { 2 }$
C) $x = 12$
D) $x = 3$

Question 59

Solve.
$$\frac { 3 } { 7 } = \log _ { e } x$$

A) $e ^ { 3 / 7 } = x$
B) $\mathrm { e } ^ { \mathrm { x } } = \frac { 3 } { 7 }$
C) $x ^ { e } = \frac { 3 } { 7 }$
D) $x ^ { 3 / 7 } = e$

Question 60

Solve.
$$\log _ { 3 } x = - 2$$

A) $x = 1$
B) $x = - 6$
C) $x = \frac { 1 } { 8 }$
D) $x = \frac { 1 } { 9 }$

Question 61

Write the expression as the sum or difference of single logarithms.
$\log _ { 9 } \left( \frac { 8 \sqrt { x } } { y } \right)$

A) $\log _ { 9 } 8 \cdot \frac { 1 } { 2 } \log _ { 9 } m \div \log _ { 9 } y$
B) $\log _ { 9 } 8 + \frac { 1 } { 2 } \log _ { 9 } x - \log _ { 9 } y$
C) $\log _ { 9 } ( 8 \sqrt { x } ) - \log _ { 9 } y$
D) $\log _ { 9 } y - \log _ { 9 } 8 - \frac { 1 } { 2 } \log _ { 9 } x$

Question 62

Graph.
$y=\log _{1 / 4^{x}}$
A)

B)
C)
D)

Question 63

A solution is an acid if its pH is less than 7 and a base if its pH is greater than 7. The pH is defined by pH = -log10[H+],
where [H+] is the concentration of hydrogen ions in the solution. Use this information to answer the question
The concentration of hydrogen ions in a solution is approximately $10 ^ { - 9.7 }$ . What is the pH of the solution? Is the solution an acid or a base?

A) The $\mathrm { pH }$ is 9.7. The solution is a base.
B) The $\mathrm { pH }$ is 9.7. The solution is an acid.
C) The $\mathrm { pH }$ is $- 9.7$ . The solution is a base.
D) The $\mathrm { pH }$ is $- 9.7$ . The solution is an acid.

Question 64

Express as a sum of logarithms.
$\log _ { 10 } x y$

A) $\log _ { 5 } x - \log _ { 5 } y$
B) $\log _ { 10 } x + \log _ { 10 } y$
C) $\log _ { 5 } x + \log _ { 5 } y$
D) $\log _ { 10 } x - \log _ { 10 } y$

Question 65

Write the expression as the sum or difference of single logarithms.
$\log _ { 13 } \sqrt { \frac { x y } { 5 } }$

A) $\frac { 1 } { 2 } \log _ { 13 } x + \frac { 1 } { 2 } \log _ { 13 } y - \log _ { 13 } 5$
B) $\frac { 1 } { 2 } \log _ { 13 } x \cdot \frac { 1 } { 2 } \log _ { 13 } y + \frac { 1 } { 2 } \log _ { 13 } 5$
C) $\frac { 1 } { 2 } \log _ { 13 } x + \frac { 1 } { 2 } \log _ { 13 } y - \frac { 1 } { 2 } \log _ { 13 } 5$
D) $\frac { 1 } { 2 } \log _ { 13 } x y - \frac { 1 } { 2 } \log _ { 13 } 5$

Question 66

Graph.
On a coordinate plane, graph the function $\mathrm { f }$ and the function $\mathrm { f } ^ { - 1 }$ . Then graph a dashed line for the equation $\mathrm { y } =$
$$f ( x ) = \log _ { 5 } x , f ^ { - 1 } ( x ) = 5 ^ { x }$$
A)
B)
C)
D)

Question 67

Write the expression as the sum or difference of single logarithms.
$\log _ { 10 } \left( \frac { \sqrt [ 3 ] { 17 } } { x ^ { 2 } y } \right)$

A) $\frac { 1 } { 3 } \log _ { 10 } 17 - 2 \log _ { 10 } x - 2 \log _ { 10 } y$
B) $\frac { 1 } { 3 } \log _ { 10 } 17 - 2 \log _ { 10 } x - \log _ { 10 } y$
C) $3 \log _ { 10 } 17 - 2 \log _ { 10 } x - \log _ { 10 } 3$
D) $\log _ { 10 } 17 - \log _ { 10 } x - \log _ { 10 } y$

Question 68

Express as a difference of logarithms.
$\log _ { 7 } a ^ { 4 }$

A) $\log ( 7 \cdot a \cdot 4 )$
B) $\log _ { 7 } a \cdot \log _ { 7 } 4$
C) $4 \log _ { 7 } a$
D) $\log 7 \cdot \log a \cdot \log 4$

Question 69

Express as a sum of logarithms.
$\log _ { 4 } ( 5 \cdot 11 )$

A) $\log 5 + \log 11 + \log 4$
B) $\log ( 5 + 11 + 4 )$
C) $\log _ { 4 } 5 + \log _ { 4 } 11$
D) $\log _ { 2 } 5 + \log _ { 2 } 11$

Question 70

A solution is an acid if its pH is less than 7 and a base if its pH is greater than 7. The pH is defined by pH = -log10[H+],
where [H+] is the concentration of hydrogen ions in the solution. Use this information to answer the question
The concentration of hydrogen ions in a solution is approximately $10 ^ { - 6.1 }$ . What is the pH of the solution? Is the solution an acid or a base?

A) The $\mathrm { pH }$ is 6.1. The solution is an acid.
B) The pH is 6.1. The solution is a base.
C) The $\mathrm { pH }$ is -6.1. The solution is a base.
D) The $\mathrm { pH }$ is -6.1. The solution is an acid.

Question 71

Express as a difference of logarithms.
$\log _ { 10 } \left( \frac { C } { E } \right)$

A) $\log C - \log E - \log 10$
B) $\log ( C - E - 10 )$
C) $\log _ { 5 } \mathrm { C } - \log _ { 5 } \mathrm { E }$
D) $\log _ { 10 } C - \log _ { 10 } E$

Question 72

A solution is an acid if its pH is less than 7 and a base if its pH is greater than 7. The pH is defined by pH = -log10[H+],
where [H+] is the concentration of hydrogen ions in the solution. Use this information to answer the question
A mixture has a pH of $6.3$ . Find the concentration of hydrogen ions $\left[ \mathrm { H } ^ { + } \right]$ in the mixture. Is the mixture an acid or a base?

A) $10 ^ { 6.3 }$ , base
B) $10 ^ { 6.3 }$ , acid
C) $10 ^ { - 6.3 }$ , acid
D) $10 ^ { - 6.3 }$ , base

Question 73

Express as a product.
$\log _ { 6 } \sqrt { z }$

A) $\frac { 1 } { 2 } \log ( 6 \cdot z )$
B) $\frac { 1 } { 2 } \log 6 + \frac { 1 } { 2 } \log z$
C) $\frac { 1 } { 2 } \log _ { 6 } z$
D) $\frac { 1 } { 2 } \log 6 \cdot \log z$

Question 74

Express as a product.
$\log _ { 4 } \frac { 13 } { 11 }$

A) $\log _ { 2 } 13 - \log _ { 2 } 11$
B) $\log _ { 4 } 13 - \log _ { 4 } 11$
C) $\log _ { 4 } 11 - \log _ { 4 } 13$
D) $\log _ { 4 } 13 + \log _ { 4 } 11$

Question 75

Express as a sum of logarithms.
$\log _ { 8 } 12 x$

A) $\log _ { 4 } 12 + \log _ { 4 } x$
B) $\log _ { 8 } 12 + \log _ { 8 } x$
C) $\log _ { 4 } 12 - \log _ { 4 } x$
D) $\log _ { 8 } 12 - \log _ { 8 } x$

Question 76

A solution is an acid if its pH is less than 7 and a base if its pH is greater than 7. The pH is defined by pH = -log10[H+],
where [H+] is the concentration of hydrogen ions in the solution. Use this information to answer the question
A mixture has a pH of 7.6. Find the concentration of hydrogen ions [ $\left. \mathrm { H } ^ { + } \right]$ in the mixture. Is the mixture an acid or a base?

A) $10 ^ { 7.6 }$ , base
B) $10 ^ { - 7.6 }$ , base
C) $10 ^ { 7.6 }$ , acid
D) $10 ^ { - 7.6 }$ , acid

Question 77

Write the expression as the sum or difference of single logarithms.
$\log _ { 5 } \frac { \sqrt [ 4 ] { x } \sqrt [ 3 ] { y } } { z ^ { 2 } }$

A) $\frac { 1 } { 4 } \log _ { 5 } x + \frac { 1 } { 3 } \log _ { 5 } y - 2 \log _ { 5 } z$
B) $\frac { 4 } { 5 } \log _ { 5 } x + \frac { 3 } { 5 } \log _ { 5 } y - \frac { 2 } { 5 } \log _ { 5 } z$
C) $\frac { 1 } { 4 } \log _ { 5 } x \cdot \frac { 1 } { 3 } \log _ { 5 } y \div 2 \log _ { 5 } z$
D) $4 \log _ { 5 } x + 3 \log _ { 5 } y - 2 \log _ { 5 } z$

Question 78

Express as a product.
$\log _ { 9 } B ^ { - 2 }$

A) $- \log _ { 9 }$ B $\cdot \log _ { 9 } 2$
B) $- \log 9 \cdot \log B \cdot \log 2$
C) $\log ( 9 \cdot B \cdot - 2 )$
D) $- 2 \log _ { 9 } B$

Question 79

Express as a difference of logarithms.
$\log _ { 6 } \left( \frac { 15 } { F } \right)$

A) $\log ( 15 - \mathrm { F } - 6 )$
B) $\log _ { 3 } 15 - \log _ { 3 } \mathrm {~F}$
C) $\log _ { 6 } 15 - \log _ { 6 } F$
D) $\log 15 - \log F - \log 6$