Question 1

Solve the exponential equation. Round to three decimal places when necessary.
-$$e ^ { - x } = 3 ^ { 2 x }$$

Accepted Answer

A)  $0.549$ 
B)  No solution 
C)  3 
D)  0

Question 2

Simplify.
-$e ^ { \ln x ^ { 42 } }$

Accepted Answer

A)  $\ln x ^ { 42 }$ 
B)  $x ^ { 42 }$ 
C)  $\mathrm { e } ^ { \mathrm { x } ^ { 42 } }$ 
D)  $x$

Question 3

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

Accepted Answer

The answer of For the function f, use composition of...

Question 4

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

Accepted Answer

The answer of Provide an appropriate response.
-Prove that the function...

Question 5

Find the logarithm using the change-of-base formula.
-$\log _ { 100 } 50$

Accepted Answer

A)  $4.6052$ 
B)  $3.912$ 
C)  $1.1772$ 
D)  $0.8495$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

The answer of For the function f, use composition of...

Question 8

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

Accepted Answer

The answer of Provide an appropriate response.
-Explain why 1 is...

Question 9

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

Accepted Answer

The answer of Provide an appropriate response.
-\[\text { Explain how...

Question 10

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

Accepted Answer

The answer of Provide an appropriate response.
-\[\text { Explain the...

Question 11

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

Accepted Answer

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 12

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

Accepted Answer

The answer of Provide an appropriate response.
-\[\text { Explain the...

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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 15

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

Accepted Answer

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 16

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

Accepted Answer

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 17

Find the logarithm using the change-of-base formula.
-$\log _ { 7.1 } 4.5$

Accepted Answer

A)  $0.6338$ 
B)  $1.3032$ 
C)  $0.7673$ 
D)  $0.6532$

Question 18

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

Accepted Answer

The answer of For the function f, use composition of...

Question 19

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

Accepted Answer

The answer of Provide an appropriate response.
-\[\text { Explain why...

Question 20

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.
-

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Exam 5: Exponential and Logarithmic Functions

Solve the exponential equation. Round to three decimal places when necessary. - $e ^ { - x } = 3 ^ { 2 x }$

Simplify. - $e ^ { \ln x ^ { 42 } }$

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

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

Find the logarithm using the change-of-base formula. - $\log _ { 100 } 50$

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

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

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

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

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

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

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

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

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

Find the logarithm using the change-of-base formula. - $\log _ { 7.1 } 4.5$

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

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

Exam 5: Exponential and Logarithmic Functions

Solve the exponential equation. Round to three decimal places when necessary. - e−x=32xe ^ { - x } = 3 ^ { 2 x }e−x=32x

Simplify. - eln⁡x42e ^ { \ln x ^ { 42 } }elnx42

For the function f, use composition of functions to show that f f−1 is as given. \mathrm { f } ^ { - 1 } \text { is as given. }f−1 is as given. - f(x)=8+xx,f−1(x)=8x−1f ( x ) = \frac { 8 + x } { x } , f ^ { - 1 } ( x ) = \frac { 8 } { x - 1 }f(x)=x8+x​,f−1(x)=x−18​

Provide an appropriate response. -Prove that the function f is not one-to-one. f(x)=2−x4f ( x ) = 2 - x ^ { 4 }f(x)=2−x4

Find the logarithm using the change-of-base formula. - log⁡10050\log _ { 100 } 50log100​50

Find the logarithm using the change-of-base formula. - log⁡2427.46\log _ { 24 } 27.46log24​27.46

For the function f, use composition of functions to show that f f−1 is as given. \mathrm { f } ^ { - 1 } \text { is as given. }f−1 is as given. - f(x)=x−28,f−1(x)=8x+2f ( x ) = \frac { x - 2 } { 8 } , f ^ { - 1 } ( x ) = 8 x + 2f(x)=8x−2​,f−1(x)=8x+2

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

Provide an appropriate response. - Explain the error in the following: log⁡68−log⁡6N=log⁡6(8−N). \text { Explain the error in the following: } \log _ { 6 } 8 - \log _ { 6 } N = \log _ { 6 } ( 8 - N ) \text {. } Explain the error in the following: log6​8−log6​N=log6​(8−N).

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

Provide an appropriate response. - Explain the error in the following: log⁡32+log⁡3M=log⁡3(2+M). \text { Explain the error in the following: } \log _ { 3 } 2 + \log _ { 3 } M = \log _ { 3 } ( 2 + M ) \text {. } Explain the error in the following: log3​2+log3​M=log3​(2+M).

Solve. -Given log⁡b2=0.5298\log _ { \mathrm { b } } 2 = 0.5298logb​2=0.5298 and log⁡b7=1.4873\log _ { \mathrm { b } } 7 = 1.4873logb​7=1.4873 , evaluate log⁡b2 b\log _ { \mathrm { b } } 2 \mathrm {~b}logb​2 b .

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

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

Express as a single logarithm and, if possible, simplify. - log⁡b8x+4(log⁡bx−log⁡by) \log _ { b } 8 x + 4 \left( \log _ { b } x - \log _ { b } y \right) logb​8x+4(logb​x−logb​y)

Find the logarithm using the change-of-base formula. - log⁡7.14.5\log _ { 7.1 } 4.5log7.1​4.5

For the function f, use composition of functions to show that f f−1 is as given. \mathrm { f } ^ { - 1 } \text { is as given. }f−1 is as given. - f(x)=x+33,f−1(x)=x3−3f ( x ) = \sqrt [ 3 ] { x + 3 } , f ^ { - 1 } ( x ) = x ^ { 3 } - 3f(x)=3x+3​,f−1(x)=x3−3

Provide an appropriate response. - Explain why log⁡213 is between 3 and 4. \text { Explain why } \log _ { 2 } 13 \text { is between } 3 \text { and } 4 \text {. } Explain why log2​13 is between 3 and 4.

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

Solve the exponential equation. Round to three decimal places when necessary. - $e ^ { - x } = 3 ^ { 2 x }$

Simplify. - $e ^ { \ln x ^ { 42 } }$

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

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

Find the logarithm using the change-of-base formula. - $\log _ { 100 } 50$

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

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

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

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

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

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

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

Find the logarithm using the change-of-base formula. - $\log _ { 7.1 } 4.5$

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

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