Question 1

Decide whether the composite functions, f ° g and g ° f, are equal to x.
-$$f ( x ) = x ^ { 2 } + 1 , g ( x ) = \sqrt { x } - 1$$

Accepted Answer

A)  No, yes 
B)  No, no 
C)  Yes, yes 
D)  Yes, no 
A)  No, yes 
B)  No, no 
C)  Yes, yes 
D)  Yes, no

Question 2

Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$$e ^ { 3 x } = 6$$

Accepted Answer

A)  $\left\{ \frac { \ln 6 } { 3 } \right\}$ 
B)  $\left\{ \frac { \ln 3 } { 6 } \right\}$ 
C)  $\{ 3 \ln 6 \}$ 
D)  $\{ 2 \mathrm { e } \}$ 
A)  $\left\{ \frac { \ln 6 } { 3 } \right\}$ 
B)  $\left\{ \frac { \ln 3 } { 6 } \right\}$ 
C)  $\{ 3 \ln 6 \}$ 
D)  $\{ 2 \mathrm { e } \}$

Question 3

Express as a single logarithm.
-$$5 \log _ { a } q - \frac { 5 } { 4 } \log _ { a } r + \frac { 1 } { 3 } \log _ { a } f - 3 \log _ { a } p$$

Accepted Answer

A)  $\log _ { a } \frac { q ^ { 5 } p ^ { 3 } } { f ^ { 1 / 3 } r ^ { 5 / 4 } }$ 
B)  $\log _ { a } \frac { q ^ { 5 } \mathrm { f } ^ { 1 / 3 } } { r ^ { 5 / 4 } p ^ { 3 } }$ 
C)  $\log _ { a } \frac { q ^ { 5 } r ^ { 5 / 4 } } { f ^ { 1 / 3 } p ^ { 3 } }$ 
D)  $\log _ { \mathrm { a } } \left( 5 \mathrm { q } - \frac { 5 } { 4 } \mathrm { r } + \frac { 1 } { 3 } \mathrm { f } - 3 \mathrm { p } \right)$ 
A)  $\log _ { a } \frac { q ^ { 5 } p ^ { 3 } } { f ^ { 1 / 3 } r ^ { 5 / 4 } }$ 
B)  $\log _ { a } \frac { q ^ { 5 } \mathrm { f } ^ { 1 / 3 } } { r ^ { 5 / 4 } p ^ { 3 } }$ 
C)  $\log _ { a } \frac { q ^ { 5 } r ^ { 5 / 4 } } { f ^ { 1 / 3 } p ^ { 3 } }$ 
D)  $\log _ { \mathrm { a } } \left( 5 \mathrm { q } - \frac { 5 } { 4 } \mathrm { r } + \frac { 1 } { 3 } \mathrm { f } - 3 \mathrm { p } \right)$

Question 4

Find the exact value of the logarithmic expression.
-$\log _ { 9 } \frac { 1 } { 729 }$

Accepted Answer

A)  3 
B)  $- 81$ 
C)  81 
D)  $- 3$ 
A)  3 
B)  $- 81$ 
C)  81 
D)  $- 3$

Question 5

Find functions f and g so that f ° g = H.
-$$H ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$$

Accepted Answer

A)  $f ( x ) = \frac { 1 } { x ^ { 2 } } ; \quad g ( x ) = - 1 / 5$ 
B)  $f ( x ) = \frac { 1 } { x } ; \quad g ( x ) = x ^ { 2 } - 5$ 
C)  $f ( x ) = x ^ { 2 } - 5 ; \quad g ( x ) = \frac { 1 } { x }$ 
D)  $f ( x ) = \frac { 1 } { x ^ { 2 } } ; \quad g ( x ) = x - 5$ 
A)  $f ( x ) = \frac { 1 } { x ^ { 2 } } ; \quad g ( x ) = - 1 / 5$ 
B)  $f ( x ) = \frac { 1 } { x } ; \quad g ( x ) = x ^ { 2 } - 5$ 
C)  $f ( x ) = x ^ { 2 } - 5 ; \quad g ( x ) = \frac { 1 } { x }$ 
D)  $f ( x ) = \frac { 1 } { x ^ { 2 } } ; \quad g ( x ) = x - 5$

Question 6

Change the exponential expression to an equivalent expression involving a logarithm.
-$$13 ^ { x } = 169$$

Accepted Answer

A)  $\log _ { 169 } 13 = x$ 
B)  $\log _ { 13 } 169 = \mathrm { x }$ 
C)  $\log _ { 169 ^ { x } } = 13$ 
D)  $\log _ { \mathrm { x } } 169 = 13$ 
A)  $\log _ { 169 } 13 = x$ 
B)  $\log _ { 13 } 169 = \mathrm { x }$ 
C)  $\log _ { 169 ^ { x } } = 13$ 
D)  $\log _ { \mathrm { x } } 169 = 13$

Question 7

Solve the equation.
-$$9 ^ { 2 x } \cdot 27 ^{( 3 - x )} = \frac { 1 } { 9 }$$

Accepted Answer

A)  $\{ - 8 \}$ 
B)  $\{ 10 \}$ 
C)  $\{ - 11 \}$ 
D)  $\left\{ \frac { 9 + \sqrt { 87 } } { 6 } , \frac { 9 - \sqrt { 87 } } { 6 } \right\}$ 
A)  $\{ - 8 \}$ 
B)  $\{ 10 \}$ 
C)  $\{ - 11 \}$ 
D)  $\left\{ \frac { 9 + \sqrt { 87 } } { 6 } , \frac { 9 - \sqrt { 87 } } { 6 } \right\}$

Question 8

The function f is one-to-one. Find its inverse.
-$$f ( x ) = ( x + 2 ) ^ { 3 } - 8$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x + 6 }$ 
B)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x - 2 } + 8$ 
C)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x + 10 }$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt [ 3 ] { \mathrm { x } + 8 } - 2$ 
A)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x + 6 }$ 
B)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x - 2 } + 8$ 
C)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { x + 10 }$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt [ 3 ] { \mathrm { x } + 8 } - 2$

Question 9

Write as the sum and/or difference of logarithms. Express powers as factors.
-$\log _ { 12 } \sqrt { \frac { \mathrm { rs } } { 13 } }$

Accepted Answer

A)  $\frac { 1 } { 2 } \log _ { 12 } \mathrm { rs } - \frac { 1 } { 2 } \log _ { 12 } 13$ 
B)  $\frac { 1 } { 2 } \log _ { 12 } \mathrm { r } + \frac { 1 } { 2 } \log _ { 12 } \mathrm {~s} - \log _ { 12 } 13$ 
C)  $\frac { 1 } { 2 } \log _ { 12 } \mathrm { r } \cdot \frac { 1 } { 2 } \log _ { 12 } \mathrm {~s} + \frac { 1 } { 2 } \log _ { 12 } 13$ 
D)  $\frac { 1 } { 2 } \log _ { 12 } \mathrm { r } + \frac { 1 } { 2 } \log _ { 12 } \mathrm {~s} - \frac { 1 } { 2 } \log _ { 12 } 13$ 
A)  $\frac { 1 } { 2 } \log _ { 12 } \mathrm { rs } - \frac { 1 } { 2 } \log _ { 12 } 13$ 
B)  $\frac { 1 } { 2 } \log _ { 12 } \mathrm { r } + \frac { 1 } { 2 } \log _ { 12 } \mathrm {~s} - \log _ { 12 } 13$ 
C)  $\frac { 1 } { 2 } \log _ { 12 } \mathrm { r } \cdot \frac { 1 } { 2 } \log _ { 12 } \mathrm {~s} + \frac { 1 } { 2 } \log _ { 12 } 13$ 
D)  $\frac { 1 } { 2 } \log _ { 12 } \mathrm { r } + \frac { 1 } { 2 } \log _ { 12 } \mathrm {~s} - \frac { 1 } { 2 } \log _ { 12 } 13$

Question 10

Choose the one alternative that best completes the statement or answers the question.
-The accompanying tables represent a function f that converts seconds to hours and a function g that converts hours to days. $$\begin{array}{l}
\begin{array} { r | r r r r r } 
\mathrm { x } & 86,400 & 172,800 & 259,200 & 345,600 & 432,000 \
\hline \mathrm { f } ( \mathrm { x } ) & 24 & 48 & 72 & 96 & 120
\end{array}\\
\begin{array} { r | r r r r r } 
\mathrm { x } & 24 & 48 & 72 & 96 & 120 \
\hline \mathrm { g } ( \mathrm { x } ) & 1 & 2 & 3 & 4 & 5
\end{array}
\end{array}$$ Express $\left( \mathrm { f } ^ { - 1 } \mathrm { gg } ^ { - 1 } \right) ( \mathrm { x } )$ symbolically.

Accepted Answer

A)  $\left( f ^ { - 1 } g ^ { - 1 } \right) ( x ) = \frac { x } { 86,400 }$ 
B)  $\left( f ^ { - 1 } \mathrm {~g} ^ { - 1 } \right) ( \mathrm { x } ) = \frac { \mathrm { x } ^ { 2 } } { 86,400 }$ 
C)  $\left( f ^ { - 1 } \mathrm {~g} ^ { - 1 } \right) ( \mathrm { x } ) = 86,400 \mathrm { x }$ 
D)  $\left( \mathrm { f } ^ { - 1 } \mathrm {~g} ^ { - 1 } \right) ( \mathrm { x } ) = 86,400 \mathrm { x } ^ { 2 }$ 
A)  $\left( f ^ { - 1 } g ^ { - 1 } \right) ( x ) = \frac { x } { 86,400 }$ 
B)  $\left( f ^ { - 1 } \mathrm {~g} ^ { - 1 } \right) ( \mathrm { x } ) = \frac { \mathrm { x } ^ { 2 } } { 86,400 }$ 
C)  $\left( f ^ { - 1 } \mathrm {~g} ^ { - 1 } \right) ( \mathrm { x } ) = 86,400 \mathrm { x }$ 
D)  $\left( \mathrm { f } ^ { - 1 } \mathrm {~g} ^ { - 1 } \right) ( \mathrm { x } ) = 86,400 \mathrm { x } ^ { 2 }$

Question 11

Decide whether or not the functions are inverses of each other.
-$$f ( x ) = 4 x + 16 , g ( x ) = \frac { 1 } { 4 } x - 4$$

Accepted Answer

A)  $Y e s$ 
B)  $\mathrm { No }$ 
A)  $Y e s$ 
B)  $\mathrm { No }$

Question 12

Write the word or phrase that best completes each statement or answers the question.
-You have two friends, Jim and Amy. Jim always yells when he speaks, and Amy always whispers. The loudness
of Jim's voice is 120 decibels, and the loudness of Amy's voice is 20 decibels. Determine how many times as
intense Jim's voice is as compared at Amy's.

Accepted Answer

Jim's voic

Question 13

Graph the function.
-$f(x)=e^{3 x}-1$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 14

Choose the one alternative that best completes the statement or answers the question.
-In 1990, the population of a country was estimated at 4 million. For any subsequent year the population, $\mathrm { P } ( \mathrm { t } )$ (in millions), can be modeled by the equation $\mathrm { P } ( \mathrm { t } ) = \frac { 240 } { 5 + 54.99 \mathrm { e } ^ { - 0.0208 \mathrm { t } } }$, where $t$ is the number of years since 1990 .
Estimate the year when the population will be 21 million.

Accepted Answer

A)  approximately the year 2088 
B)  approximately the year 2016 
C)  approximately the year 2093 
D)  approximately the year 2041 
A)  approximately the year 2088 
B)  approximately the year 2016 
C)  approximately the year 2093 
D)  approximately the year 2041

Question 15

Change the logarithmic expression to an equivalent expression involving an exponent.
-$$\log _ { 5 } 25 = x$$

Accepted Answer

A)  $5 ^ { x } = 25$ 
B)  $25 ^ { \mathrm { x } } = 5$ 
C)  $x ^ { 5 } = 25$ 
D)  $25 ^ { 5 } = \mathrm { x }$ 
A)  $5 ^ { x } = 25$ 
B)  $25 ^ { \mathrm { x } } = 5$ 
C)  $x ^ { 5 } = 25$ 
D)  $25 ^ { 5 } = \mathrm { x }$

Question 16

Write the word or phrase that best completes each statement or answers the question.
-The rates of death (in number of deaths per 100,000 population) for $20 - 24$ year olds in the United States between 1985-1993 are given below. (Source: NCHS Data Warehouse)
$\begin{array}{c|c}
\text { Year } & \text { Rate of Death } \
\hline 1985 & 134.9 \
\hline 1987 & 154.7 \
\hline 1989 & 162.9 \
\hline 1991 & 174.5 \
\hline 1993 & 182.2
\end{array}$
 A logarithmic equation that models this data is $y = 57.76 + 48.56 \ln x$ where $x$ represents the number of years sinc 1980 and y represents the rate of death in that year. Use this equation to predict the year in which the rate of death for $20 - 24$ year olds first exceeds 200 .

Accepted Answer

The answer of Write the word or phrase that best...

Question 17

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

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 18

Change the logarithmic expression to an equivalent expression involving an exponent.
-$\log _ { 1024 } 256 = \frac { 4 } { 5 }$

Accepted Answer

A)  $256 ^ { 4 / 5 } = 1024$ 
B)  $256 ^ { 5 / 4 } = 1024$ 
C)  $\left( \frac { 4 } { 5 } \right) ^ { 1024 } = 256$ 
D)  $1024^{4 / 5} = 256$ 
A)  $256 ^ { 4 / 5 } = 1024$ 
B)  $256 ^ { 5 / 4 } = 1024$ 
C)  $\left( \frac { 4 } { 5 } \right) ^ { 1024 } = 256$ 
D)  $1024^{4 / 5} = 256$

Question 19

Choose the one alternative that best completes the statement or answers the question.
-$$e ^ { x } = 13$$

Accepted Answer

A)  $\log _ { 13 } x = e$ 
B)  $\ln 13 = x$ 
C)  $\log _ { \mathrm { x } } \mathrm { e } = 13$ 
D)  $\ln x = 13$ 
A)  $\log _ { 13 } x = e$ 
B)  $\ln 13 = x$ 
C)  $\log _ { \mathrm { x } } \mathrm { e } = 13$ 
D)  $\ln x = 13$

Question 20

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

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Decide whether the composite functions, f ° g and g ° f, are equal to x. - $f ( x ) = x ^ { 2 } + 1 , g ( x ) = \sqrt { x } - 1$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $e ^ { 3 x } = 6$

Express as a single logarithm. - $5 \log _ { a } q - \frac { 5 } { 4 } \log _ { a } r + \frac { 1 } { 3 } \log _ { a } f - 3 \log _ { a } p$

Find the exact value of the logarithmic expression. - $\log _ { 9 } \frac { 1 } { 729 }$

Find functions f and g so that f ° g = H. - $H ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$

Change the exponential expression to an equivalent expression involving a logarithm. - $13 ^ { x } = 169$

Solve the equation. - $9 ^ { 2 x } \cdot 27 ^{( 3 - x )} = \frac { 1 } { 9 }$

The function f is one-to-one. Find its inverse. - $f ( x ) = ( x + 2 ) ^ { 3 } - 8$

Write as the sum and/or difference of logarithms. Express powers as factors. - $\log _ { 12 } \sqrt { \frac { \mathrm { rs } } { 13 } }$

Decide whether or not the functions are inverses of each other. - $f ( x ) = 4 x + 16 , g ( x ) = \frac { 1 } { 4 } x - 4$

Graph the function. - $f(x)=e^{3 x}-1$ $Graph the function. - f(x)=e^{3 x}-1$

Change the logarithmic expression to an equivalent expression involving an exponent. - $\log _ { 5 } 25 = x$

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

Change the logarithmic expression to an equivalent expression involving an exponent. - $\log _ { 1024 } 256 = \frac { 4 } { 5 }$

Choose the one alternative that best completes the statement or answers the question. - $e ^ { x } = 13$

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

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 4: Exponential and Logarithmic Functions

Decide whether the composite functions, f ° g and g ° f, are equal to x. - f(x)=x2+1,g(x)=x−1f ( x ) = x ^ { 2 } + 1 , g ( x ) = \sqrt { x } - 1f(x)=x2+1,g(x)=x​−1

Solve the exponential equation. Express the solution set in terms of natural logarithms. - e3x=6e ^ { 3 x } = 6e3x=6

Express as a single logarithm. - 5log⁡aq−54log⁡ar+13log⁡af−3log⁡ap5 \log _ { a } q - \frac { 5 } { 4 } \log _ { a } r + \frac { 1 } { 3 } \log _ { a } f - 3 \log _ { a } p5loga​q−45​loga​r+31​loga​f−3loga​p

Find the exact value of the logarithmic expression. - log⁡91729\log _ { 9 } \frac { 1 } { 729 }log9​7291​

Find functions f and g so that f ° g = H. - H(x)=1x2−5H ( x ) = \frac { 1 } { x ^ { 2 } - 5 }H(x)=x2−51​

Change the exponential expression to an equivalent expression involving a logarithm. - 13x=16913 ^ { x } = 16913x=169

Solve the equation. - 92x⋅27(3−x)=199 ^ { 2 x } \cdot 27 ^{( 3 - x )} = \frac { 1 } { 9 }92x⋅27(3−x)=91​

The function f is one-to-one. Find its inverse. - f(x)=(x+2)3−8f ( x ) = ( x + 2 ) ^ { 3 } - 8f(x)=(x+2)3−8

Write as the sum and/or difference of logarithms. Express powers as factors. - log⁡12rs13\log _ { 12 } \sqrt { \frac { \mathrm { rs } } { 13 } }log12​13rs​​

Decide whether or not the functions are inverses of each other. - f(x)=4x+16,g(x)=14x−4f ( x ) = 4 x + 16 , g ( x ) = \frac { 1 } { 4 } x - 4f(x)=4x+16,g(x)=41​x−4

Graph the function. - f(x)=e3x−1f(x)=e^{3 x}-1f(x)=e3x−1

Change the logarithmic expression to an equivalent expression involving an exponent. - log⁡525=x\log _ { 5 } 25 = xlog5​25=x

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

Change the logarithmic expression to an equivalent expression involving an exponent. - log⁡1024256=45\log _ { 1024 } 256 = \frac { 4 } { 5 }log1024​256=54​

Choose the one alternative that best completes the statement or answers the question. - ex=13e ^ { x } = 13ex=13

Graph the function. - y=log⁡2xy=\log _{2} xy=log2​x

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Decide whether the composite functions, f ° g and g ° f, are equal to x. - $f ( x ) = x ^ { 2 } + 1 , g ( x ) = \sqrt { x } - 1$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $e ^ { 3 x } = 6$

Express as a single logarithm. - $5 \log _ { a } q - \frac { 5 } { 4 } \log _ { a } r + \frac { 1 } { 3 } \log _ { a } f - 3 \log _ { a } p$

Find the exact value of the logarithmic expression. - $\log _ { 9 } \frac { 1 } { 729 }$

Find functions f and g so that f ° g = H. - $H ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$

Change the exponential expression to an equivalent expression involving a logarithm. - $13 ^ { x } = 169$

Solve the equation. - $9 ^ { 2 x } \cdot 27 ^{( 3 - x )} = \frac { 1 } { 9 }$

The function f is one-to-one. Find its inverse. - $f ( x ) = ( x + 2 ) ^ { 3 } - 8$

Write as the sum and/or difference of logarithms. Express powers as factors. - $\log _ { 12 } \sqrt { \frac { \mathrm { rs } } { 13 } }$

Decide whether or not the functions are inverses of each other. - $f ( x ) = 4 x + 16 , g ( x ) = \frac { 1 } { 4 } x - 4$

Graph the function. - $f(x)=e^{3 x}-1$ $Graph the function. - f(x)=e^{3 x}-1$

Change the logarithmic expression to an equivalent expression involving an exponent. - $\log _ { 5 } 25 = x$

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

Change the logarithmic expression to an equivalent expression involving an exponent. - $\log _ { 1024 } 256 = \frac { 4 } { 5 }$

Choose the one alternative that best completes the statement or answers the question. - $e ^ { x } = 13$

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