Question 1

Solve the problem.
-Given that $f ( x ) = 5 \ln 2 x$, find $f ^ { - 1 } ( x )$ and give the domain and range of $f ^ { - 1 } ( x )$.

Accepted Answer

A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { x - 5 }$, domain $= ( - \infty , \infty )$, range $= ( 0 , \infty )$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } } / 5$, domain $= ( - \infty , \infty )$, range $= ( 0 , \infty )$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } } / 5$, domain $= ( 0 , \infty )$, range $= ( - \infty , \infty )$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } - 5 }$, domain $= ( 0 , \infty )$, range $= ( - \infty , \infty )$ 
A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { x - 5 }$, domain $= ( - \infty , \infty )$, range $= ( 0 , \infty )$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } } / 5$, domain $= ( - \infty , \infty )$, range $= ( 0 , \infty )$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } } / 5$, domain $= ( 0 , \infty )$, range $= ( - \infty , \infty )$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } - 5 }$, domain $= ( 0 , \infty )$, range $= ( - \infty , \infty )$

Question 2

Graph the function. Give the domain and range.
-

Accepted Answer

A)  domain: $( - \infty , 2 )$; range: $( - \infty , \infty )$
  
B)  domain: $( - 2 , \infty )$; range: $( - \infty , \infty )$
  
C)  domain: $( - \infty , 2 )$; range: $( - \infty , \infty )$
  
D)  domain: $( - 2 , \infty )$; range: $( - \infty , \infty )$
  
A)  domain: $( - \infty , 2 )$; range: $( - \infty , \infty )$
  
B)  domain: $( - 2 , \infty )$; range: $( - \infty , \infty )$
  
C)  domain: $( - \infty , 2 )$; range: $( - \infty , \infty )$
  
D)  domain: $( - 2 , \infty )$; range: $( - \infty , \infty )$

Question 3

Solve the equation and express the solution in exact form.
-$$\ln ( - x ) + \ln 4 = \ln ( 3 x - 9 )$$

Accepted Answer

A)  $\left\{ - \frac { 9 } { 7 } \right\}$ 
B)  $\left\{ \frac { 7 } { 9 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { 9 } { 7 } \right\}$ 
A)  $\left\{ - \frac { 9 } { 7 } \right\}$ 
B)  $\left\{ \frac { 7 } { 9 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { 9 } { 7 } \right\}$

Question 4

Use properties of logarithms to evaluate the expression.
-$\log _ { 10 } ( 0.0001 ) ^ { 5 }$

Accepted Answer

A)  $- 625$ 
B)  $- 20$ 
C)  $\frac { 1 } { 20 }$ 
D)  $\log 625$ 
A)  $- 625$ 
B)  $- 20$ 
C)  $\frac { 1 } { 20 }$ 
D)  $\log 625$

Question 5

Solve the problem.
-Let $\mathrm { u } = \ln \mathrm { a }$ and $\mathrm { v } = \ln \mathrm { b }$. Write the following expression in terms of $\mathrm { u }$ and $\mathrm { v }$ without using the function $\ln$.
$\ln \frac { a ^ { 7 } } { b ^ { 2 } }$

Accepted Answer

A)  $7 u + 2 v$ 
B)  $u ^ { 7 } - v ^ { 2 }$ 
C)  $7 \mathrm { u } - 2 \mathrm { v }$ 
D)  $2 \mathrm { u } - 7 \mathrm { v }$ 
A)  $7 u + 2 v$ 
B)  $u ^ { 7 } - v ^ { 2 }$ 
C)  $7 \mathrm { u } - 2 \mathrm { v }$ 
D)  $2 \mathrm { u } - 7 \mathrm { v }$

Question 6

Solve the problem.
-Which of the following is the same as $\log ( 12 x ) - \log ( 3 x )$ for $x > 0$ ?

Accepted Answer

A)  $\log ( 9 x )$ 
B)  $4 \log x$ 
C)  $\frac { \log ( 12 x ) } { \log ( 3 x ) }$ 
D)  $\log ( 4 )$ 
A)  $\log ( 9 x )$ 
B)  $4 \log x$ 
C)  $\frac { \log ( 12 x ) } { \log ( 3 x ) }$ 
D)  $\log ( 4 )$

Question 7

Solve the equation and express the solution in exact form.
-$\log _ { 2 } \sqrt { 2 x ^ { 2 } } = \frac { 9 } { 2 }$

Accepted Answer

A)  $\{ 16 \}$ 
B)  $\{ - 16,16 \}$ 
C)  $\{ - 256,256 \}$ 
D)  $\{ - 8,8 \}$ 
A)  $\{ 16 \}$ 
B)  $\{ - 16,16 \}$ 
C)  $\{ - 256,256 \}$ 
D)  $\{ - 8,8 \}$

Question 8

Solve the problem.
-What is the rate on an investment that triples $4336 in 27 years? Assume interest is compounded monthly.

Accepted Answer

A)  3.4% 
B)  5.1% 
C)  4.1% 
D)  2.7% 
A)  3.4% 
B)  5.1% 
C)  4.1% 
D)  2.7%

Question 9

Find the function value. If the result is irrational, round your answer to the nearest thousandth.
-Let f(x) = 2x. Find f(6)

Accepted Answer

A)  12 
B)  128 
C)  36 
D)  64 
A)  12 
B)  128 
C)  36 
D)  64

Question 10

Find the domain and range of the inverse of the given function.
-$$f ( x ) = x ^ { 3 } + 4$$

Accepted Answer

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

Question 11

Evaluate the logarithm.
-$\log _ { 12 } ( - 1 )$

Accepted Answer

A)  1 
B)  0 
C)  Undefined 
D)  $- 1$ 
A)  1 
B)  0 
C)  Undefined 
D)  $- 1$

Question 12

Write the word or phrase that best completes each statement or answers the question.
-$$\text { Write the algebraic equation which can be used to find the exact solution of }$$

Accepted Answer

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

Question 13

Solve the problem.
-The growth in population of a city can be seen using the formula $\mathrm { p } ( \mathrm { t } ) = 8338 \mathrm { e } ^ { 0.006 \mathrm { t } }$ where t is the number of years. Use this formula to calculate the population after 10 years.

Accepted Answer

A)  17,707 
B)  26,561 
C)  9401 
D)  8854 
A)  17,707 
B)  26,561 
C)  9401 
D)  8854

Question 14

Determine whether or not the function is one-to-one.
-

Accepted Answer

A)  No 
B)  Yes 
A)  No 
B)  Yes

Question 15

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

Accepted Answer

A)  $\{ - 1 \}$ 
B)  $\left\{ \frac { 1 } { 9 } \right\}$ 
C)  $\{ - 6 \}$ 
D)  $\left\{ \frac { 1 } { 8 } \right\}$ 
A)  $\{ - 1 \}$ 
B)  $\left\{ \frac { 1 } { 9 } \right\}$ 
C)  $\{ - 6 \}$ 
D)  $\left\{ \frac { 1 } { 8 } \right\}$

Question 16

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth.
-2(x - 1) = 13

Accepted Answer

A)  {7.50} 
B)  {4.70} 
C)  {3.91} 
D)  {2.70} 
A)  {7.50} 
B)  {4.70} 
C)  {3.91} 
D)  {2.70}

Question 17

Solve the problem.
-Suppose $f ( x ) = 28.4 + 1.3 \log ( x + 1 )$ models salinity of ocean water to depths of 1000 meters at a certain latitude. $x$ is the depth in meters and $f ( x )$ is in grams of salt per kilogram of seawater. Approximate the depth (to the nearest tenth of a meter) where the salinity equals 29 .

Accepted Answer

A)  $1.9 \mathrm {~m}$ 
B)  $- 0.9 \mathrm {~m}$ 
C)  $- 0.7 \mathrm {~m}$ 
D)  $3.9 \mathrm {~m}$ 
A)  $1.9 \mathrm {~m}$ 
B)  $- 0.9 \mathrm {~m}$ 
C)  $- 0.7 \mathrm {~m}$ 
D)  $3.9 \mathrm {~m}$

Question 18

Find the value. Give an approximation to four decimal places.
-$\ln \left( 8.48 \times e ^ { 6 } \right)$

Accepted Answer

A)  $8.1377$ 
B)  $2.1377$ 
C)  $6.0208$ 
D)  $- 3.8623$ 
A)  $8.1377$ 
B)  $2.1377$ 
C)  $6.0208$ 
D)  $- 3.8623$

Question 19

Solve the equation. Round to the nearest thousandth.
-$$23 ( 3 - x ) = 30$$

Accepted Answer

A)  $\{ - 1.696 \}$ 
B)  $\{ 1.915 \}$ 
C)  $\{ 1.696 \}$ 
D)  $\{ 3.922 \}$ 
A)  $\{ - 1.696 \}$ 
B)  $\{ 1.915 \}$ 
C)  $\{ 1.696 \}$ 
D)  $\{ 3.922 \}$

Question 20

Solve the problem.
-The growth in the mouse population at a certain county dump can be modeled by the exponential function A(t)= 967e0.021t, where t is the number of months since the population was first recorded.
Estimate the population after 35 months.

Accepted Answer

A)  2059 
B)  988 
C)  1009 
D)  2017 
A)  2059 
B)  988 
C)  1009 
D)  2017

Solve the problem. -Given that $f ( x ) = 5 \ln 2 x$ , find $f ^ { - 1 } ( x )$ and give the domain and range of $f ^ { - 1 } ( x )$ .

Graph the function. Give the domain and range. -

Solve the equation and express the solution in exact form. - $\ln ( - x ) + \ln 4 = \ln ( 3 x - 9 )$

Use properties of logarithms to evaluate the expression. - $\log _ { 10 } ( 0.0001 ) ^ { 5 }$

Solve the problem. -Let $\mathrm { u } = \ln \mathrm { a }$ and $\mathrm { v } = \ln \mathrm { b }$ . Write the following expression in terms of $\mathrm { u }$ and $\mathrm { v }$ without using the function $\ln$ . $\ln \frac { a ^ { 7 } } { b ^ { 2 } }$

Solve the problem. -Which of the following is the same as $\log ( 12 x ) - \log ( 3 x )$ for $x > 0$ ?

Solve the equation and express the solution in exact form. - $\log _ { 2 } \sqrt { 2 x ^ { 2 } } = \frac { 9 } { 2 }$

Solve the problem. -What is the rate on an investment that triples $4336 in 27 years? Assume interest is compounded monthly.

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let f(x) = 2x. Find f(6)

Find the domain and range of the inverse of the given function. - $f ( x ) = x ^ { 3 } + 4$

Evaluate the logarithm. - $\log _ { 12 } ( - 1 )$

Write the word or phrase that best completes each statement or answers the question. - $\text { Write the algebraic equation which can be used to find the exact solution of }$

Solve the problem. -The growth in population of a city can be seen using the formula $\mathrm { p } ( \mathrm { t } ) = 8338 \mathrm { e } ^ { 0.006 \mathrm { t } }$ where t is the number of years. Use this formula to calculate the population after 10 years.

Determine whether or not the function is one-to-one. -

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

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. -2(x - 1) = 13

Find the value. Give an approximation to four decimal places. - $\ln \left( 8.48 \times e ^ { 6 } \right)$

Solve the equation. Round to the nearest thousandth. - $23 ( 3 - x ) = 30$

Solve the problem. -The growth in the mouse population at a certain county dump can be modeled by the exponential function A(t)= 967e0.021t, where t is the number of months since the population was first recorded. Estimate the population after 35 months.

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Exam 4: Inverse, Exponential, and Logarithmic Functions

Solve the problem. -Given that f(x)=5ln⁡2xf ( x ) = 5 \ln 2 xf(x)=5ln2x , find f−1(x)f ^ { - 1 } ( x )f−1(x) and give the domain and range of f−1(x)f ^ { - 1 } ( x )f−1(x) .