Question 1

Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$$7 ^ { x } = 6 ^ { x + 7 }$$

Accepted Answer

A)  $81.36$ 
B)  $3.36$ 
C)  $42.00$ 
D)  $40.68$ 
A)  $81.36$ 
B)  $3.36$ 
C)  $42.00$ 
D)  $40.68$

Question 2

Approximate the number using a calculator. Round your answer to three decimal places.
-$\mathrm { e } ^ { - 1.5 }$

Accepted Answer

A)  $0.223$ 
B)  $- 4.077$ 
C)  $- 0.223$ 
D)  $0.523$ 
A)  $0.223$ 
B)  $- 4.077$ 
C)  $- 0.223$ 
D)  $0.523$

Question 3

The graph of a logarithmic function is given. Select the function for the graph from the options.
-

Accepted Answer

A)  $f ( x ) = - \log _ { 4 } x$ 
B)  $f ( x ) = \log _ { 4 } ( - x )$ 
C)  $f ( x ) = 1 - \log _ { 4 } x$ 
D)  $f ( x ) = \log _ { 4 } x$ 
A)  $f ( x ) = - \log _ { 4 } x$ 
B)  $f ( x ) = \log _ { 4 } ( - x )$ 
C)  $f ( x ) = 1 - \log _ { 4 } x$ 
D)  $f ( x ) = \log _ { 4 } x$

Question 4

The graph of a logarithmic function is given. Select the function for the graph from the options.
-

Accepted Answer

A)  $f ( x ) = 1 - \log _ { 4 } x$ 
B)  $f ( x ) = \log _ { 4 } ( - x )$ 
C)  $f ( x ) = - \log _ { 4 } x$ 
D)  $f ( x ) = \log _ { 4 } x$ 
A)  $f ( x ) = 1 - \log _ { 4 } x$ 
B)  $f ( x ) = \log _ { 4 } ( - x )$ 
C)  $f ( x ) = - \log _ { 4 } x$ 
D)  $f ( x ) = \log _ { 4 } x$

Question 5

Solve.
-The value of a particular investment follows a pattern of exponential growth. In the year 2000 , you invested money in a money market account. The value of your investment t years after 2000 is given by the exponential growth model $\mathrm { A } = 1700 \mathrm { e } ^ { 0.062 \mathrm { t } }$. When will the account be worth $\$ 2466 ?$

Accepted Answer

A)  2006 
B)  2007 
C)  2008 
D)  2005 
A)  2006 
B)  2007 
C)  2008 
D)  2005

Question 6

Write the equation in its equivalent logarithmic form.
-$$43 = 64$$

Accepted Answer

A)  $\log _ { 4 } 64 = 3$ 
B)  $\log _ { 64 } 4 = 3$ 
C)  $\log _ { 3 } 64 = 4$ 
D)  $\log _ { 4 } 3 = 64$ 
A)  $\log _ { 4 } 64 = 3$ 
B)  $\log _ { 64 } 4 = 3$ 
C)  $\log _ { 3 } 64 = 4$ 
D)  $\log _ { 4 } 3 = 64$

Question 7

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm
whose coefficient is 1. Where possible, evaluate logarithmic expressions.
-$$\log _ { \mathrm { C } } \mathrm { x } + \log _ { \mathrm { c } } \mathrm { y }$$

Accepted Answer

A)  $\log _ { \mathrm { C } } ( \mathrm { xy } )$ 
B)  $\log _ { \mathrm { C } } \mathrm { x } \cdot \log _ { \mathrm { C } } \mathrm { y }$ 
C)  $\log _ { \mathrm { C } } ( \mathrm { x } + \mathrm { y } )$ 
D)  $\log _ { \mathrm { c } } \left( \frac { \mathrm { x } } { \mathrm { y } } \right)$ 
A)  $\log _ { \mathrm { C } } ( \mathrm { xy } )$ 
B)  $\log _ { \mathrm { C } } \mathrm { x } \cdot \log _ { \mathrm { C } } \mathrm { y }$ 
C)  $\log _ { \mathrm { C } } ( \mathrm { x } + \mathrm { y } )$ 
D)  $\log _ { \mathrm { c } } \left( \frac { \mathrm { x } } { \mathrm { y } } \right)$

Question 8

Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$$10 ^ { x } = 4.04$$

Accepted Answer

A)  $0.61$ 
B)  $10,964.78$ 
C)  $1.4$ 
D)  $40.4$ 
A)  $0.61$ 
B)  $10,964.78$ 
C)  $1.4$ 
D)  $40.4$

Question 9

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$$\log _ { 2 } ( x - 2 ) - \log _ { 2 } ( x - 3 ) = 4$$

Accepted Answer

A)  $\left\{ \frac { 46 } { 15 } \right\}$ 
B)  $\left\{ \frac { 1 } { 15 } \right\}$ 
C)  $\left\{ - \frac { 46 } { 15 } \right\}$ 
D)  $\varnothing$ ) 
A)  $\left\{ \frac { 46 } { 15 } \right\}$ 
B)  $\left\{ \frac { 1 } { 15 } \right\}$ 
C)  $\left\{ - \frac { 46 } { 15 } \right\}$ 
D)  $\varnothing$ )

Question 10

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log _ { 5 } ( 7 \cdot 3 )$

Accepted Answer

A)  $\log _ { 5 } 7 + \log _ { 5 } 3$ 
B)  $\log _ { 5 } 7 - \log _ { 5 } 3$ 
C)  $\log _ { 5 } 21$ 
D)  $\left( \log _ { 5 } 7 \right) \left( \log _ { 5 } 3 \right)$ 
A)  $\log _ { 5 } 7 + \log _ { 5 } 3$ 
B)  $\log _ { 5 } 7 - \log _ { 5 } 3$ 
C)  $\log _ { 5 } 21$ 
D)  $\left( \log _ { 5 } 7 \right) \left( \log _ { 5 } 3 \right)$

Question 11

Evaluate the expression without using a calculator.
-$\mathrm { e } ^ { \ln 126 }$

Accepted Answer

A)  126 
B)  $- 126$ 
C)  $\mathrm { e } ^ { 126 }$ 
D)  $\ln 126$ 
A)  126 
B)  $- 126$ 
C)  $\mathrm { e } ^ { 126 }$ 
D)  $\ln 126$

Question 12

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$\log _ { 35 } ( x + 70 ) = 3 - \log _ { 35 } x$

Accepted Answer

A)  $\{ 175 \}$ 
B)  $\{ - 175 \}$ 
C)  $\{ 245 \}$ 
D)  $\{ - 245 \}$ 
A)  $\{ 175 \}$ 
B)  $\{ - 175 \}$ 
C)  $\{ 245 \}$ 
D)  $\{ - 245 \}$

Question 13

Evaluate the expression without using a calculator.
-$\log _ { 11 } 1$

Accepted Answer

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

Question 14

Write the equation in its equivalent exponential form.
-$\log _ { 2 } 8 = 3$

Accepted Answer

A)  $23 = 8$ 
B)  $32 = 8$ 
C)  $28 = 3$ 
D)  $83 = 2$ 
A)  $23 = 8$ 
B)  $32 = 8$ 
C)  $28 = 3$ 
D)  $83 = 2$

Question 15

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$\log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( x - 4 )$

Accepted Answer

A)  $\{ 7 \}$ 
B)  $\{ 3 \}$ 
C)  $\{ - 7 \}$ 
D)  $\{ - 3 \}$ 
A)  $\{ 7 \}$ 
B)  $\{ 3 \}$ 
C)  $\{ - 7 \}$ 
D)  $\{ - 3 \}$

Question 16

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$$\log _ { 8 } 7 + \log _ { 8 } x = 1$$

Accepted Answer

A)  $\left\{ \frac { 8 } { 7 } \right\}$ 
B)  $\left\{ \frac { 7 } { 8 } \right\}$ 
C)  $\{ \sqrt [ 7 ] { 8 } \}$ 
D)  $\left\{ \frac { 1 } { 7 } \right\}$ ){ 
A)  $\left\{ \frac { 8 } { 7 } \right\}$ 
B)  $\left\{ \frac { 7 } { 8 } \right\}$ 
C)  $\{ \sqrt [ 7 ] { 8 } \}$ 
D)  $\left\{ \frac { 1 } { 7 } \right\}$ ){

Question 17

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$\ln ( x - 6 ) + \ln ( x + 1 ) = \ln ( x - 15 )$

Accepted Answer

A)  $\{ - 3 \}$ 
B)  $\{ 3 \}$ 
C)  $\{ 3 , - 3 \}$ 
D)  $\varnothing$ 
A)  $\{ - 3 \}$ 
B)  $\{ 3 \}$ 
C)  $\{ 3 , - 3 \}$ 
D)  $\varnothing$

Question 18

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\ln \sqrt { \frac { x } { y } }$

Accepted Answer

A)  $\frac { 1 } { 2 } \ln x - \frac { 1 } { 2 } \ln y$ 
B)  $\ln \sqrt { x } - \ln \sqrt { y }$ 
C)  $\frac { 1 } { 2 } \ln \frac { x } { y }$ 
D)  $\frac { 1 } { 2 } \ln x - \ln y$ 
A)  $\frac { 1 } { 2 } \ln x - \frac { 1 } { 2 } \ln y$ 
B)  $\ln \sqrt { x } - \ln \sqrt { y }$ 
C)  $\frac { 1 } { 2 } \ln \frac { x } { y }$ 
D)  $\frac { 1 } { 2 } \ln x - \ln y$

Question 19

Evaluate the expression without using a calculator.
-$\log _ { 5 } 25$

Accepted Answer

A)  2 
B)  10 
C)  $\frac { 1 } { 2 }$ 
D)  1 
A)  2 
B)  10 
C)  $\frac { 1 } { 2 }$ 
D)  1

Question 20

Choose the one alternative that best completes the statement or answers the question.
Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm, and then round to three
decimal places.
-$$y = 13 ( 5.4 ) ^ { x }$$

Accepted Answer

A)  $y = 13 \mathrm { e } ^ { \mathrm { x } \ln 5.4 } , \mathrm { y } = 13 \mathrm { e } ^ { 1.686 \mathrm { x } }$ 
B)  $y = 5.4 \mathrm { e } ^ { \mathrm { x } } \ln 13 , \mathrm { y } = 5.4 \mathrm { e } ^ { 2.565 \mathrm { x } }$ 
C)  $y = 13 \mathrm { e } ^ { 5.4 \mathrm { x } } , \mathrm { y } = 132.718 ^ { 1.686 \mathrm { x } }$ 
D)  $\mathrm { y } = ( \ln 13 ) \mathrm { e } ^ { \mathrm { x } } \ln 5.4 , \mathrm { y } = 2.565 \mathrm { e } ^ { 1.686 \mathrm { x } }$ 
A)  $y = 13 \mathrm { e } ^ { \mathrm { x } \ln 5.4 } , \mathrm { y } = 13 \mathrm { e } ^ { 1.686 \mathrm { x } }$ 
B)  $y = 5.4 \mathrm { e } ^ { \mathrm { x } } \ln 13 , \mathrm { y } = 5.4 \mathrm { e } ^ { 2.565 \mathrm { x } }$ 
C)  $y = 13 \mathrm { e } ^ { 5.4 \mathrm { x } } , \mathrm { y } = 132.718 ^ { 1.686 \mathrm { x } }$ 
D)  $\mathrm { y } = ( \ln 13 ) \mathrm { e } ^ { \mathrm { x } } \ln 5.4 , \mathrm { y } = 2.565 \mathrm { e } ^ { 1.686 \mathrm { x } }$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $7 ^ { x } = 6 ^ { x + 7 }$

Approximate the number using a calculator. Round your answer to three decimal places. - $\mathrm { e } ^ { - 1.5 }$

The graph of a logarithmic function is given. Select the function for the graph from the options. -

The graph of a logarithmic function is given. Select the function for the graph from the options. -

Write the equation in its equivalent logarithmic form. - $43 = 64$

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\log _ { \mathrm { C } } \mathrm { x } + \log _ { \mathrm { c } } \mathrm { y }$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $10 ^ { x } = 4.04$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 2 } ( x - 2 ) - \log _ { 2 } ( x - 3 ) = 4$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 5 } ( 7 \cdot 3 )$

Evaluate the expression without using a calculator. - $\mathrm { e } ^ { \ln 126 }$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 35 } ( x + 70 ) = 3 - \log _ { 35 } x$

Evaluate the expression without using a calculator. - $\log _ { 11 } 1$

Write the equation in its equivalent exponential form. - $\log _ { 2 } 8 = 3$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( x - 4 )$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 8 } 7 + \log _ { 8 } x = 1$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\ln ( x - 6 ) + \ln ( x + 1 ) = \ln ( x - 15 )$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\ln \sqrt { \frac { x } { y } }$

Evaluate the expression without using a calculator. - $\log _ { 5 } 25$

Choose the one alternative that best completes the statement or answers the question. Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm, and then round to three decimal places. - $y = 13 ( 5.4 ) ^ { x }$

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Exam 4: Exponential and Logarithmic Functions

Solve the exponential equation. Express the solution set in terms of natural logarithms. - 7x=6x+77 ^ { x } = 6 ^ { x + 7 }7x=6x+7

Approximate the number using a calculator. Round your answer to three decimal places. - e−1.5\mathrm { e } ^ { - 1.5 }e−1.5

The graph of a logarithmic function is given. Select the function for the graph from the options. -

The graph of a logarithmic function is given. Select the function for the graph from the options. -

Write the equation in its equivalent logarithmic form. - 43=6443 = 6443=64

Solve the exponential equation. Express the solution set in terms of natural logarithms. - 10x=4.0410 ^ { x } = 4.0410x=4.04

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - log⁡2(x−2)−log⁡2(x−3)=4\log _ { 2 } ( x - 2 ) - \log _ { 2 } ( x - 3 ) = 4log2​(x−2)−log2​(x−3)=4

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - log⁡5(7⋅3)\log _ { 5 } ( 7 \cdot 3 )log5​(7⋅3)

Evaluate the expression without using a calculator. - eln⁡126\mathrm { e } ^ { \ln 126 }eln126

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - log⁡35(x+70)=3−log⁡35x\log _ { 35 } ( x + 70 ) = 3 - \log _ { 35 } xlog35​(x+70)=3−log35​x

Evaluate the expression without using a calculator. - log⁡111\log _ { 11 } 1log11​1

Write the equation in its equivalent exponential form. - log⁡28=3\log _ { 2 } 8 = 3log2​8=3

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - log⁡2(x−1)=1+log⁡2(x−4)\log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( x - 4 )log2​(x−1)=1+log2​(x−4)

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - log⁡87+log⁡8x=1\log _ { 8 } 7 + \log _ { 8 } x = 1log8​7+log8​x=1

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - ln⁡(x−6)+ln⁡(x+1)=ln⁡(x−15)\ln ( x - 6 ) + \ln ( x + 1 ) = \ln ( x - 15 )ln(x−6)+ln(x+1)=ln(x−15)

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - ln⁡xy\ln \sqrt { \frac { x } { y } }lnyx​​

Evaluate the expression without using a calculator. - log⁡525\log _ { 5 } 25log5​25

Choose the one alternative that best completes the statement or answers the question. Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm, and then round to three decimal places. - y=13(5.4)xy = 13 ( 5.4 ) ^ { x }y=13(5.4)x

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $7 ^ { x } = 6 ^ { x + 7 }$

Approximate the number using a calculator. Round your answer to three decimal places. - $\mathrm { e } ^ { - 1.5 }$

Write the equation in its equivalent logarithmic form. - $43 = 64$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $10 ^ { x } = 4.04$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 2 } ( x - 2 ) - \log _ { 2 } ( x - 3 ) = 4$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 5 } ( 7 \cdot 3 )$

Evaluate the expression without using a calculator. - $\mathrm { e } ^ { \ln 126 }$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 35 } ( x + 70 ) = 3 - \log _ { 35 } x$

Evaluate the expression without using a calculator. - $\log _ { 11 } 1$

Write the equation in its equivalent exponential form. - $\log _ { 2 } 8 = 3$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( x - 4 )$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 8 } 7 + \log _ { 8 } x = 1$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\ln ( x - 6 ) + \ln ( x + 1 ) = \ln ( x - 15 )$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\ln \sqrt { \frac { x } { y } }$

Evaluate the expression without using a calculator. - $\log _ { 5 } 25$

Choose the one alternative that best completes the statement or answers the question. Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm, and then round to three decimal places. - $y = 13 ( 5.4 ) ^ { x }$