Deck 5: Exponential and Logarithmic Functions

Question 1

Match the function with one of the graphs.
$$f ( x ) = e ^ { x } - 6$$

A)

B)

C)

D)

Answer 1

C

Question 2

For the function f, use composition of functions to show that f $\mathrm { f } ^ { - 1 } \text { is as given. }$
$$f ( x ) = \frac { x - 2 } { 8 } , f ^ { - 1 } ( x ) = 8 x + 2$$

Answer 2

Answers may vary. One possible solution is: 1. $\left( f ^ { - 1 } \right.$ of $) ( x ) = f ^ { - 1 } ( f ( x ) ) = f ^ { - 1 } ( ( x - 2 ) / 8 ) = 8 ( ( x - 2 ) / 8 ) + 2 = ( x - 2 ) + 2 = x$ ;
2. $\left( f \circ f ^ { - 1 } \right) ( x ) = f \left( f ^ { - 1 } ( x ) \right) = f ( 8 x + 2 ) = ( ( 8 x + 2 ) - 2 ) / 8 = 8 x / 8 = x$

Question 3

Match the function with one of the graphs.
$$f ( x ) = e ^ { x + 2 }$$

A)

B)

C)

D)

Answer 3

D

Question 4

Choose the function that might be used as a model for the data in the scatter plot.

A) Logarithmic, $f ( x ) = a + b \ln x$
B) Exponential, $\mathrm { f } ( \mathrm { x } ) = \mathrm { ab } ^ { \mathrm { x } }$ or $\mathrm { f } ( \mathrm { x } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { \mathrm { kx } } , \mathrm { k } > 0$
C) Quadratic, $f ( x ) = a x ^ { 2 } + b x + c$
D) Exponential, $\mathrm { f } ( \mathrm { x } ) = \mathrm { ab } ^ { - \mathrm { x } }$ or $\mathrm { f } ( \mathrm { x } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { - \mathrm { kx } } , \mathrm { k } > 0$

Question 5

Choose the function that might be used as a model for the data in the scatter plot.

A) Logarithmic, $f ( x ) = a + b \ln x$
B) Exponential, $\mathrm { f } ( \mathrm { x } ) = a b ^ { - \mathrm { x } }$ or $\mathrm { f } ( \mathrm { x } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { - \mathrm { kx } } , \mathrm { k } > 0$
C) Polynomial, not quadratic
D) Logistic, $\mathrm { f } ( \mathrm { x } ) = \frac { \mathrm { a } } { 1 + b \mathrm { e } ^ { - \mathrm { kx } } }$

Question 6

Choose the function that might be used as a model for the data in the scatter plot.

A) Logistic, $f ( x ) = \frac { a } { 1 + b e ^ { - k x } }$
B) Exponential, $\mathrm { f } ( \mathrm { x } ) = a b ^ { - \mathrm { x } }$ or $\mathrm { f } ( \mathrm { x } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { - \mathrm { kx } } , \mathrm { k } > 0$
C) Quadratic, $f ( x ) = a x ^ { 2 } + b x + c$
D) Polynomial, not quadratic

Question 7

Match the function with one of the graphs.
$$f ( x ) = 3 ^ { x } + 4$$

A)

B)

C)

D)

Question 8

Provide an appropriate response.
Without using a calculator, determine which of these numbers is larger: $$\sqrt { 8 ^ { 3 } } \text { or } 8 \sqrt { 3 } \text {. }$$

Question 9

Choose the function that might be used as a model for the data in the scatter plot.

A) Polynomial, not quadratic
B) Logarithmic, $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } + \mathrm { b } \ln \mathrm { x }$
C) Quadratic, $f ( x ) = a x ^ { 2 } + b x + c$
D) Exponential, $\mathrm { f } ( \mathrm { x } ) = \mathrm { ab } ^ { \mathrm { x } }$ or $\mathrm { f } ( \mathrm { x } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { \mathrm { kx } } , \mathrm { k } > 0$

Question 10

Match the function with one of the graphs.
$$f ( x ) = 2 e ^ { - x }$$

A)

B)

C)

D)

Question 11

Match the function with one of the graphs.
$$f ( x ) = 5 + e ^ { x }$$

A)

B)

C)

D)

Question 12

Provide an appropriate response.
$$\text { Explain why } f ( x ) = 2 ^ { x } \text { is an exponential function but } f ( x ) = x ^ { 2 } \text { is not. }$$

Question 13

Choose the function that might be used as a model for the data in the scatter plot.

A) Logarithmic, $f ( x ) = a + b \ln x$
B) Polynomial, not quadratic
C) Exponential, $\mathrm { f } ( \mathrm { x } ) = \mathrm { ab } { } ^ { - \mathrm { x } }$ or $\mathrm { f } ( \mathrm { x } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { - \mathrm { kx } } , \mathrm { k } > 0$
D) Exponential, $\mathrm { f } ( \mathrm { x } ) = \mathrm { ab } ^ { \mathrm { x } }$ or $\mathrm { f } ( \mathrm { x } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { \mathrm { kx } } , \mathrm { k } > 0$

Question 14

Provide an appropriate response.
$$\text { Explain the error in the following: } \log _ { 6 } 8 - \log _ { 6 } N = \log _ { 6 } ( 8 - N ) \text {. }$$

Question 15

Match the function with one of the graphs.
$$f ( x ) = - \left( \frac { 1 } { 3 } \right) ^ { x }$$

A)

B)

C)

D)

Question 16

Choose the function that might be used as a model for the data in the scatter plot.

A) Polynomial, not quadratic
B) Exponential, $\mathrm { f } ( \mathrm { x } ) = a \mathrm {~b} ^ { \mathrm { x } }$ or $\mathrm { f } ( \mathrm { x } ) = \mathrm { P } _ { 0 } \mathrm { e } ^ { \mathrm { kx } } , \mathrm { k } > 0$
C) Logistic, $f ( x ) = \frac { a } { 1 + b e ^ { - k x } }$
D) Logarithmic, $f ( x ) = a + b \ln x$

Question 17

Provide an appropriate response.
$$\text { Explain the error in the following: } \log _ { 3 } 2 + \log _ { 3 } M = \log _ { 3 } ( 2 + M ) \text {. }$$

Question 18

Provide an appropriate response.
$$\text { Explain how the graph of } f ( x ) = e ^ { x } \text { could be used to graph the function } g ( x ) = 1 + \ln x \text {. }$$

Question 19

For the function f, use composition of functions to show that f $\mathrm { f } ^ { - 1 } \text { is as given. }$
$$f ( x ) = x ^ { 3 } - 4 , f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x + 4 }$$

Question 20

Provide an appropriate response.
Prove that the function f is one-to-one. $$f ( x ) = x ^ { 3 } - 4$$

Question 21

Provide an appropriate response.
Prove that the function f is not one-to-one. $f ( x ) = | x | - 2$

Question 22

Provide an appropriate response.
Explain the error in the following: $$\log _ { 4 } 3 y = \log _ { 4 } 3 \cdot \log _ { 4 } y$$

Question 23

For the function f, use composition of functions to show that f $\mathrm { f } ^ { - 1 } \text { is as given. }$
$$f ( x ) = \sqrt [ 3 ] { x + 3 } , f ^ { - 1 } ( x ) = x ^ { 3 } - 3$$

Question 24

For the function f, use composition of functions to show that f $\mathrm { f } ^ { - 1 } \text { is as given. }$
$$f ( x ) = \frac { 7 } { 4 } x , f ^ { - 1 } ( x ) = \frac { 4 } { 7 } x$$

Question 25

Provide an appropriate response.
Without using a calculator, determine which of these numbers is larger: $$\pi ^ { 1.3 } \text { or } \pi ^ { 2.4 }$$

Question 26

Provide an appropriate response.
Prove that the function f is one-to-one. $$f ( x ) = 6 - \frac { 1 } { 2 } x$$

Question 27

For the function f, use composition of functions to show that f $$\mathrm { f } ^ { - 1 } \text { is as given. }$$
$$f ( x ) = \frac { 8 + x } { x } , f ^ { - 1 } ( x ) = \frac { 8 } { x - 1 }$$

Question 28

For the function f, use composition of functions to show that f $$\mathrm { f } ^ { - 1 } \text { is as given. }$$
$$f ( x ) = \frac { x + 4 } { 6 } , f ^ { - 1 } ( x ) = 6 x - 4$$

Question 29

Provide an appropriate response.
$$\text { Explain why } \log _ { 2 } 13 \text { is between } 3 \text { and } 4 \text {. }$$

Question 30

Provide an appropriate response.
Prove that the function f is not one-to-one. $f ( x ) = 2 - x ^ { 4 }$

Question 31

Provide an appropriate response.
Suppose that $1000 is invested for 5 years at 4% interest, compounded annually. In what
year will the most interest be earned? Why?

Question 32

For the function f, use composition of functions to show that f $$\mathrm { f } ^ { - 1 } \text { is as given. }$$
$$f ( x ) = - \frac { 5 } { 3 } x , f ^ { - 1 } ( x ) = - \frac { 3 } { 5 } x$$

Question 33

Provide an appropriate response.
$$\text { Without using a calculator, explain why } 2 \pi \text { must be greater than } 8 \text { but less than } 16 \text {. }$$

Question 34

Provide an appropriate response.
$$\text { Explain how the graph of } f ( x ) = \ln x \text { could be used to graph the function } g ( x ) = e ^ { x - 1 }$$

Question 35

Provide an appropriate response.
$$\text { Explain how the equation } \log x = 1 \text { could be solved using the graph of } f ( x ) = \log x \text {. }$$

Question 36

Provide an appropriate response.
Prove that the function f is one-to-one. $$f ( x ) = 3 x - 1$$

Question 37

Provide an appropriate response.
Explain why 1 is excluded from being a logarithmic base.

Question 38

For the function f, use composition of functions to show that f $$\mathrm { f } ^ { - 1 } \text { is as given. }$$
$$f ( x ) = \frac { x + 7 } { 2 } , f ^ { - 1 } = 2 x - 7$$

Question 39

Provide an appropriate response.
Prove that the function f is not one-to-one. $$f ( x ) = \frac { 2 } { x ^ { 6 } }$$

Question 40

Provide an appropriate response.
Prove that the function f is not one-to-one. $$f ( x ) = x ^ { 2 } + 2$$

Question 41

Provide an appropriate response.
Prove that the function f is one-to-one. $$f ( x ) = \sqrt [ 3 ] { x } + 5$$

Question 42

Using the horizontal-line test, determine whether the function is one-to-one.
$f(x)=\frac{7}{x^{2}-2}$

A) Yes
B) $\mathrm { No }$

Question 43

Find the logarithm using the change-of-base formula.
$\log _ { 24 } 27.46$

A) $1.4387$
B) $1.0424$
C) $0.9593$
D) $1.1442$

Question 44

Provide an appropriate response.
The product, power, and quotient rules enable us to simplify expressions like $\log _ { \mathrm { a } } \left( \frac { \mathrm { rs } } { \mathrm { pv } } \right)$ . Explain why such expressions can always be simplified without using the quotient rule.

Question 45

Find the inverse of the relation.
$$\{ ( - 8,6 ) , ( - 12,6 ) , ( 5 , - 14 ) \}$$

A) $\{ ( - 8 , - 2 ) , ( - 8 , - 12 ) , ( - 14,5 ) \}$
B) $\{ ( 6 , - 8 ) , ( 5 , - 12 ) , ( - 14 , - 2 ) \}$
C) $\{ ( 6 , - 8 ) , ( - 2 , - 12 ) , ( - 14,5 ) \}$
D) $\{ ( 6 , - 8 ) , ( 6 , - 12 ) , ( - 14,5 ) \}$

Question 46

Provide an appropriate response.
Explain why negative numbers do not have logarithms.

Question 47

Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic logarithmic function.
$$f ( x ) = \frac { 1 } { 4 } \log ( x + 2 ) + 4$$

A) Shift $y = \log x$ to the left 2 units,
stretch it vertically, and shift up 4 units

B) Shift $y = \log x$ to the right 2 units
,shrink it vertically, and shift up 4 units

C) Shift $y = \log x$ to the left 2 units,
shrink it vertically, and shift down 4 units

D) Shift $y = \log x$ to the left 2 units,
shrink it vertically, and shift up 4 units

Question 48

Find the domain and range of the inverse of the given function.
$$f ( x ) = \sqrt { x - 4 }$$

A) Domain: $[ 4 , \infty )$ ; range: $[ 0 , \infty )$
B) Domain and range: all real numbers
C) Domain: $[ 0 , \infty )$ ; range: $[ 4 , \infty )$
D) Domain: all real numbers; range: $[ 4 , \infty )$

Question 49

Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic logarithmic function.
A) Shift $y = \ln x$ right 2 units

B) Shift $y = \ln x$ left 2 units

C) Shift $y = \ln x$ right 2 units

D) Shift $y = \ln x$ left 2 units

Question 50

Graph the function.
$f(x)=5(4 x-2)$

A)

B)

C)

D)

Question 51

Provide an appropriate response.
$$\text { Find } f \left( f ^ { - 1 } ( 1598 ) \right) \text { given that } f ( x ) = \sqrt [ 5 ] { \frac { 6 x - 2 } { 5 x - 3 } } \text {. }$$

Question 52

Solve.
In 1985, the number of female athletes participating in Summer Olympic-Type Games was 450. In 1996, about 3650 participated in the Summer Olympics in Atlanta. Assuming that P(0)= 500 and
That the exponential model applies, find the value of k rounded to the hundredths place, and write
The function. A) $\mathrm { k } = 0.19 ; \mathrm { P } ( \mathrm { t } ) = 500 \mathrm { e } ^ { 0.29 \mathrm { t } }$
B) $\mathrm { k } = 0.17 ; \mathrm { P } ( \mathrm { t } ) = 500 \mathrm { e } ^ { 0.17 \mathrm { t } }$
C) $\mathrm { k } = 0.19 ; \mathrm { P } ( \mathrm { t } ) = 500 \mathrm { e } ^ { 0.19 \mathrm { t } }$
D) $\mathrm { k } = 0.21 ; \mathrm { P } ( \mathrm { t } ) = 500 \mathrm { e } ^ { 0.21 \mathrm { t } }$

Question 53

Simplify.
$$\log _ { b } \sqrt { b ^ { 5 } }$$

A) 5
B) $\frac { 5 } { 2 }$
C) 3
D) $\frac { 1 } { 2 }$

Question 54

Solve the logarithmic equation.
$\log _ { 3 } ( 8 x - 6 ) = 3$

A) $\frac { \log _ { 3 } 3 + 6 } { 8 }$
B) $\frac { 11 } { 3 }$
C) $\frac { 33 } { 8 }$
D) 25

Question 55

Express as a single logarithm and, if possible, simplify.
$\log _ { b } 8 x + 4 \left( \log _ { b } x - \log _ { b } y \right)$

A) $\log _ { b } \left( 8 x + x ^ { 4 } - y ^ { 4 } \right)$
B) $\frac { \left( \log _ { b } 8 x \right) \left( \log _ { b } x ^ { 4 } \right) } { \log \mathrm { y } ^ { 4 } }$
C) $\log _ { b } \frac { 8 x ^ { 5 } } { y ^ { 4 } }$
D) $\log b \frac { 32 x ^ { 2 } } { y ^ { 4 } }$

Question 56

Solve.
How long will it take for $\$ 3700$ to grow to $\$ 34,300$ at an interest rate of $10.3 \%$ if the interest is compounded continuously? Round the number of years to the nearest hundredth.

A) $0.22 \mathrm { yr }$
B) $2.16 \mathrm { yr }$
C) $2161.95 \mathrm { yr }$
D) $21.62 \mathrm { yr }$

Question 57

Convert to an exponential equation.
$\log _ { 8 } 64 = \mathrm { t }$

A) $64 ^ { t } = 8$
B) $8 ^ { t } = 64$
C) $864 = t$
D) $t ^ { 8 } = 64$

Question 58

Solve the logarithmic equation.
$$\log _ { 16 } x = \frac { 1 } { 2 }$$

A) $0.00001526$
B) 256
C) 65,536
D) 4

Question 59

Find the domain and the vertical asymptote of the function.
$$g ( x ) = \ln ( x - 8 )$$

A) Domain: $( 8 , \infty )$ ; vertical asymptote: $x = 8$
B) Domain: $( - \infty , \infty )$ ; vertical asymptote: none
C) Domain: $( - 8 , \infty )$ ; vertical asymptote: $x = - 8$
D) Domain: $( 0 , \infty )$ ; vertical asymptote: $x = 0$

Question 60

Determine whether the given function is one-to-one. If it is one-to-one, find a formula for the inverse.
$$f ( x ) = \frac { 2 } { x + 3 }$$

A) $f ^ { - 1 } ( x ) = \frac { - 3 x + 2 } { x }$
B) $f ^ { - 1 } ( x ) = \frac { x } { 3 + 2 x }$
C) $f ^ { - 1 } ( x ) = \frac { 3 + 2 x } { x }$
D) Not one-to-one

Question 61

Find the following using a calculator. Round to four decimal places.
$\log 93,500$

A) $4.9708$
B) $6.8407$
C) $6.8408$
D) $6.8404$

Question 62

Express as a difference of logarithms.
$\log _ { 4 } \frac { 9 } { 11 }$

A) $\log _ { 4 } 9 \div \log _ { 4 } 11$
B) $\log _ { 4 } 9 - \log _ { 4 } 11$
C) $\log _ { 4 } 11 - \log _ { 4 } 9$
D) $\log _ { 2 } 9 - \log _ { 2 } 11$

Question 63

Graph the piecewise function.

Question 64

Solve.
Given $\log _ { \mathrm { b } } 2 = 0.5298$ and $\log _ { \mathrm { b } } 7 = 1.4873$ , evaluate $\log _ { \mathrm { b } } 2 \mathrm {~b}$ .

A) $2.0171$
B) $0.5298$
C) $1.0757$
D) $1.5298$

Question 65

Evaluate to four decimal places using a calculator.
$\mathrm { e } ^ { 1.233 }$

A) $7.3891$
B) $3.4315$
C) $3.3516$
D) $1.5203$

Question 66

Solve the problem.
Suppose the amount of a radioactive element remaining in a sample of 100 milligrams after $x$ years can be described by $\mathrm { A } ( \mathrm { x } ) = 100 \mathrm { e } ^ { - 0.01569 \mathrm { x } }$ . How much is remaining after 58 years? Round the answer to the nearest hundredth of a milligram.

A) $0.40 \mathrm { mg }$
B) $248.44 \mathrm { mg }$
C) $91.00 \mathrm { mg }$
D) $40.25 \mathrm { mg }$

Question 67

Convert to a logarithmic equation.
$$4 ^ { 2 } = 16$$

A) $4 = \log _ { 2 } 16$
B) $2 = \log _ { 4 } 16$
C) $2 = \log _ { 16 } 4$
D) $16 = \log _ { 4 } 2$

Question 68

Graph the function.
$$f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$$

A)

B)

C)

D)

Question 69

Convert to a logarithmic equation.
$10 ^ { 0.6021 } = 4$

A) $0.6021 = \log _ { 9 } 10$
B) $4 = \log _ { 10 } 0.6021$
C) $10 = \log _ { 9 } 0.6021$
D) $0.6021 = \log _ { 10 } 4$

Question 70

Graph the function.
$f(x)=4(x-3)+3$

A)

B)


C)

D)

Question 71

Graph the function.
$$y = \frac { 1 } { 6 } e ^ { x }$$

A)

B)

C)

D)

Question 72

Graph the function. Describe its position relative to the graph of the indicated basic function.
A) Shrunk vertically

B) Shrunk horizontally

C) Shrunk vertically

D) Shrunk horizontally

Question 73

The graph of a one-to-one function f is given. Sketch the graph of the inverse function $$\mathrm { f } ^ { - 1 }$$ , on the same set of axes. Use a
dashed line for the inverse.

A)

B)

C)

D)

Question 74

Find the domain and range of the inverse of the given function.
$$f ( x ) = x ^ { 2 } - 8 ; x \geq 0$$

A) Domain: $[ - 8 , \infty )$ ; range: all real numbers
B) Domain and range: all real numbers
C) Domain: $[ 0 , \infty )$ ; range: $[ - 8 , \infty )$
D) Domain: $[ - 8 , \infty )$ ; range: $[ 0 , \infty )$

Question 75

Find the domain and the vertical asymptote of the function.
$$f ( x ) = - 2 - 4 \log ( x + 1 )$$

A) Domain $( - \infty , - 1 )$ ; vertical asymptote: $x = - 1$
B) Domain $( - \infty , 1 )$ ; vertical asymptote: $x = 1$
C) Domain $( - 1 , \infty )$ ; vertical asymptote: $x = - 1$
D) Domain $( 1 , \infty )$ ; vertical asymptote: $x = 1$

Question 76

Graph the piecewise function.
$$f ( x ) = \left\{ \begin{array} { l l } e ^ { - x } - 8 , & \text { for } x < - 2 \\x - 2 , & \text { for } - 2 \leq x < 1 \\x ^ { 3 } , & \text { for } x \geq 1\end{array} \right.$$

A)

B)

C)

D)

Question 77

Solve the problem.
An initial investment of $14,000 is appreciated for 4 years in an account that earns 13% interest, compounded semiannually. Find the amount of money in the account at the end of the period.

A)$22,826.63
B)$23,169.94
C)$9169.94
D)$21,755.81

Question 78

Solve.
Find the hydrogen ion concentration of a solution whose $\mathrm { pH }$ is $6.8$ . Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ .

A) $1.58 \times 10 ^ { - 7 }$
B) $6.31 \times 106$
C) $- 0.8325089$
D) $0.83250891$

Question 79

Solve the problem.
In September 1998 the population of the country of West Goma in millions was modeled by $\mathrm { f } ( \mathrm { x } ) = 16.6 \mathrm { e } ^ { 0.0012 \mathrm { x } }$ . At the same time the population of East Goma in millions was modeled by $\mathrm { g } ( \mathrm { x } ) = 13.4 \mathrm { e } ^ { 0.0133 \mathrm { x } }$ . In both formulas $\mathrm { x }$ is the year, where $\mathrm { x } = 0$ corresponds to September $1998 .$ Assuming these trends continue, estimate the year when the population of West Goma will equal the population of East Goma.

A) 1980
B) 18
C) 2013
D) 2016

Question 80

Find the domain and the vertical asymptote of the function.
$$f ( x ) = \log ( x - 10 )$$

A) Domain $( 0 , \infty )$ ; vertical asymptote: $x = 0$
B) Domain $( 10 , \infty )$ ; vertical asymptote: $x = 10$
C) Domain: $( - 10 , \infty )$ ; vertical asymptote: $x = - 10$
D) Domain $( 1 , \infty )$ ; vertical asymptote: $x = 1$