Question 1

Find the domain of the logarithmic function.
-$$f ( x ) = \log \left( x ^ { 2 } - 8 x + 12 \right)$$

Accepted Answer

A)  $( - \infty , 2 ) \cup ( 6 , \infty )$ 
B)  $( - 2,6 )$ 
C)  $( 6 , \infty )$ 
D)  $( - \infty , - 2 )$ ( 
A)  $( - \infty , 2 ) \cup ( 6 , \infty )$ 
B)  $( - 2,6 )$ 
C)  $( 6 , \infty )$ 
D)  $( - \infty , - 2 )$ (

Question 2

Approximate the number using a calculator. Round your answer to three decimal places.
-$\mathrm { e } ^ { - 2.1 }$

Accepted Answer

A)  $0.122$ 
B)  $- 5.708$ 
C)  $- 0.122$ 
D)  $0.422$ 
A)  $0.122$ 
B)  $- 5.708$ 
C)  $- 0.122$ 
D)  $0.422$

Question 3

Model Exponential Growth and Decay
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 A = 9400e0.061t. By what percentage is the account increasing each year?

Accepted Answer

A) 6.1% 
B) 6.5% 
C) 6.7% 
D) 6.8% 
A) 6.1% 
B) 6.5% 
C) 6.7% 
D) 6.8%

Question 4

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log _ { 4 } \left( \frac { x ^ { 2 } } { y ^ { 6 } } \right)$

Accepted Answer

A)  $2 \log _ { 4 } x - 6 \log _ { 4 } y$ 
B)  $2 \log _ { 4 } x + 6 \log _ { 4 } y$ 
C)  $\frac { 1 } { 3 } \log _ { 4 } \left( \frac { \mathrm { x } } { \mathrm { y } } \right)$ 
D)  $6 \log _ { 4 } y - 2 \log _ { 4 } x$ 
A)  $2 \log _ { 4 } x - 6 \log _ { 4 } y$ 
B)  $2 \log _ { 4 } x + 6 \log _ { 4 } y$ 
C)  $\frac { 1 } { 3 } \log _ { 4 } \left( \frac { \mathrm { x } } { \mathrm { y } } \right)$ 
D)  $6 \log _ { 4 } y - 2 \log _ { 4 } x$

Question 5

Use the One-to-One Property of Logarithms to Solve Logarithmic Equations
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.
-$$\ln ( x - 5 ) - \ln ( x + 7 ) = \ln ( x - 6 ) - \ln ( x + 1 )$$

Accepted Answer

A)  $\left\{ \frac { 37 } { 5 } \right\}$ 
B)  $\left\{ - \frac { 37 } { 5 } \right\}$ 
C)  $\left\{ \frac { 47 } { 5 } \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \frac { 37 } { 5 } \right\}$ 
B)  $\left\{ - \frac { 37 } { 5 } \right\}$ 
C)  $\left\{ \frac { 47 } { 5 } \right\}$ 
D)  $\varnothing$

Question 6

Use the Quotient Rule
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\ln \left( \frac { e ^ { 3 } } { 5 } \right)$

Accepted Answer

A)  $3 - \ln 5$ 
B)  $\ln e ^ { 3 } - \ln 5$ 
C)  $3 + \ln 5$ 
D)  $\ln e ^ { 3 } + \ln 5$ 
A)  $3 - \ln 5$ 
B)  $\ln e ^ { 3 } - \ln 5$ 
C)  $3 + \ln 5$ 
D)  $\ln e ^ { 3 } + \ln 5$

Question 7

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
-$\log _ { 0.4 } 17$

Accepted Answer

A)  $- 3.0920$ 
B)  $- 0.3234$ 
C)  $0.8325$ 
D)  $1.6284$ 
A)  $- 3.0920$ 
B)  $- 0.3234$ 
C)  $0.8325$ 
D)  $1.6284$

Question 8

Solve the problem.
-The logistic growth function $f ( t ) = \frac { 720 } { 1 + 9.3 e ^ { - 0.12 t } }$ describes the population of a species of butterflies t months after they are introduced to a non-threatening habitat. How many butterflies were initially introduced to the habitat?

Accepted Answer

A)  70 butterflies 
B)  720 butterflies 
C)  9 butterflies 
D)  2 butterflies 
A)  70 butterflies 
B)  720 butterflies 
C)  9 butterflies 
D)  2 butterflies

Question 9

Solve the equation by expressing each side as a power of the same base and then equating exponents.
-$$3 x = 27$$

Accepted Answer

A)  $\{ 3 \}$ 
B)  $\{ 9 \}$ 
C)  $\{ 4 \}$ 
D)  $\{ 2 \}$ 
A)  $\{ 3 \}$ 
B)  $\{ 9 \}$ 
C)  $\{ 4 \}$ 
D)  $\{ 2 \}$

Question 10

Use the Product Rule
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log _ { 6 } ( 7 \cdot 3 )$

Accepted Answer

A)  $\log _ { 6 } 7 + \log _ { 6 } 3$ 
B)  $\log _ { 6 } 7 - \log _ { 6 } 3$ 
C)  $\log _ { 6 } 21$ 
D)  $\left( \log _ { 6 } 7 \right) \left( \log _ { 6 } 3 \right)$ 
A)  $\log _ { 6 } 7 + \log _ { 6 } 3$ 
B)  $\log _ { 6 } 7 - \log _ { 6 } 3$ 
C)  $\log _ { 6 } 21$ 
D)  $\left( \log _ { 6 } 7 \right) \left( \log _ { 6 } 3 \right)$

Question 11

Solve the problem.
-The population of a certain country is growing at a rate of $1.2 \%$ per year. How long will it take for this country's population to double? Use the formula $t = \frac { \ln 2 } { k }$, which gives the time, $t$, for a population with growth rate $k$, to double. (Round to the nearest whole year.)

Accepted Answer

A)  58 years 
B)  59 years 
C)  60 years 
D)  57 years 
A)  58 years 
B)  59 years 
C)  60 years 
D)  57 years

Question 12

Use the Definition of a Logarithm to Solve Logarithmic Equations
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.
-$$\log _ { 6 } ( x - 1 ) = - 3$$

Accepted Answer

A)  $\left\{ \frac { 217 } { 216 } \right\}$ 
B)  $\left\{ - \frac { 215 } { 216 } \right\}$ 
C)  $\left\{ \frac { 217 } { 729 } \right\}$ 
D)  $\left\{ - \frac { 215 } { 729 } \right\}$ 
A)  $\left\{ \frac { 217 } { 216 } \right\}$ 
B)  $\left\{ - \frac { 215 } { 216 } \right\}$ 
C)  $\left\{ \frac { 217 } { 729 } \right\}$ 
D)  $\left\{ - \frac { 215 } { 729 } \right\}$

Question 13

Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$$3 ^ { 9 x } = 2.3$$

Accepted Answer

A)  $\left\{ \frac { \ln 2.3 } { 9 \ln 3 } \right\}$ 
B)  $\left\{ \frac { \ln 2.3 } { 3 \ln 9 } \right\}$ 
C)  $\left\{ \frac { 9 \ln 2.3 } { \ln 3 } \right\}$ 
D)  $\left\{ \frac { 2.3 \ln 9 } { \ln 3 } \right\}$ 
A)  $\left\{ \frac { \ln 2.3 } { 9 \ln 3 } \right\}$ 
B)  $\left\{ \frac { \ln 2.3 } { 3 \ln 9 } \right\}$ 
C)  $\left\{ \frac { 9 \ln 2.3 } { \ln 3 } \right\}$ 
D)  $\left\{ \frac { 2.3 \ln 9 } { \ln 3 } \right\}$

Question 14

Graph the function.
-Use the graph of $\log _ { 5 } x$ to obtain the graph of $f ( x ) = \frac { 1 } { 2 } \log _ { 5 } x$.

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  Graph the functions in the same rectangular coordinate system. 
A) 
  
B) 
  
C) 
  
D) 
  Graph the functions in the same rectangular coordinate system.

Question 15

Evaluate the expression without using a calculator.
-$$\log _ { 4 } \frac { 1 } { 16 }$$

Accepted Answer

A)  $- 2$ 
B)  8 
C)  $\frac { 1 } { 2 }$ 
D)  2 
A)  $- 2$ 
B)  8 
C)  $\frac { 1 } { 2 }$ 
D)  2

Question 16

Solve the problem.
-The function $f ( x ) = 1 + 1.5 \ln ( x + 1 )$ models the average number of free-throws a basketball player can make consecutively during practice as a function of time, where $x$ is the number of consecutive days the basketball player has practiced for two hours. After how many days of practice can the basketball player make an average of 6 consecutive free throws?

Accepted Answer

A)  27 days 
B)  29 days 
C)  105 days 
D)  107 days 
A)  27 days 
B)  29 days 
C)  105 days 
D)  107 days

Question 17

Use the Definition of a Logarithm to Solve Logarithmic Equations
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.
-$$\ln \sqrt { x + 8 } = 7$$

Accepted Answer

A)  $\left\{ \mathrm { e } ^ { 14 } - 8 \right\}$ 
B)  $\left\{ \mathrm { e } ^ { 14 } + 8 \right\}$ 
C)  $\left\{ \frac { \mathrm { e } ^ { 7 } } { 2 } + 8 \right\}$ 
D)  $\left\{ e ^ { 7 } - 8 \right\}$ 
A)  $\left\{ \mathrm { e } ^ { 14 } - 8 \right\}$ 
B)  $\left\{ \mathrm { e } ^ { 14 } + 8 \right\}$ 
C)  $\left\{ \frac { \mathrm { e } ^ { 7 } } { 2 } + 8 \right\}$ 
D)  $\left\{ e ^ { 7 } - 8 \right\}$

Question 18

Graph the function.
-Use the graph of $f ( x ) = \ln x$ to obtain the graph of $g ( x ) = 4 - \ln x$.

Accepted Answer

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

Question 19

Find the domain of the logarithmic function.
-$$f ( x ) = \ln ( 6 - x )$$

Accepted Answer

A)  $( - \infty , 6 )$ 
B)  $( - \infty , 0 )$ 
C)  $( - 6 , \infty )$ 
D)  $( - \infty , 6 )$ or $( 6 , \infty )$ 
A)  $( - \infty , 6 )$ 
B)  $( - \infty , 0 )$ 
C)  $( - 6 , \infty )$ 
D)  $( - \infty , 6 )$ or $( 6 , \infty )$

Question 20

Graph the function.
-Use the graph of $f ( x ) = \ln x$ to obtain the graph of $g ( x ) = 3 \ln x$.

Accepted Answer

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

Find the domain of the logarithmic function. - $f ( x ) = \log \left( x ^ { 2 } - 8 x + 12 \right)$

Approximate the number using a calculator. Round your answer to three decimal places. - $\mathrm { e } ^ { - 2.1 }$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 4 } \left( \frac { x ^ { 2 } } { y ^ { 6 } } \right)$

Use the One-to-One Property of Logarithms to Solve Logarithmic Equations 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. - $\ln ( x - 5 ) - \ln ( x + 7 ) = \ln ( x - 6 ) - \ln ( x + 1 )$

Use the Quotient Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\ln \left( \frac { e ^ { 3 } } { 5 } \right)$

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places - $\log _ { 0.4 } 17$

Solve the problem. -The logistic growth function $f ( t ) = \frac { 720 } { 1 + 9.3 e ^ { - 0.12 t } }$ describes the population of a species of butterflies t months after they are introduced to a non-threatening habitat. How many butterflies were initially introduced to the habitat?

Solve the equation by expressing each side as a power of the same base and then equating exponents. - $3 x = 27$

Use the Product Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 6 } ( 7 \cdot 3 )$

Use the Definition of a Logarithm to Solve Logarithmic Equations 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. - $\log _ { 6 } ( x - 1 ) = - 3$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $3 ^ { 9 x } = 2.3$

Graph the function. -Use the graph of $\log _ { 5 } x$ to obtain the graph of $f ( x ) = \frac { 1 } { 2 } \log _ { 5 } x$ . $Graph the function. -Use the graph of \log _ { 5 } x to obtain the graph of f ( x ) = \frac { 1 } { 2 } \log _ { 5 } x .$

Evaluate the expression without using a calculator. - $\log _ { 4 } \frac { 1 } { 16 }$

Use the Definition of a Logarithm to Solve Logarithmic Equations 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. - $\ln \sqrt { x + 8 } = 7$

Graph the function. -Use the graph of $f ( x ) = \ln x$ to obtain the graph of $g ( x ) = 4 - \ln x$ . $Graph the function. -Use the graph of f ( x ) = \ln x to obtain the graph of g ( x ) = 4 - \ln x .$

Find the domain of the logarithmic function. - $f ( x ) = \ln ( 6 - x )$

Graph the function. -Use the graph of $f ( x ) = \ln x$ to obtain the graph of $g ( x ) = 3 \ln x$ . $Graph the function. -Use the graph of f ( x ) = \ln x to obtain the graph of g ( x ) = 3 \ln x .$

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections

Sequences, Induction, and Probability

Filters

Exam 4: Exponential and Logarithmic Functions

Find the domain of the logarithmic function. - f(x)=log⁡(x2−8x+12)f ( x ) = \log \left( x ^ { 2 } - 8 x + 12 \right)f(x)=log(x2−8x+12)

Approximate the number using a calculator. Round your answer to three decimal places. - e−2.1\mathrm { e } ^ { - 2.1 }e−2.1

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - log⁡4(x2y6)\log _ { 4 } \left( \frac { x ^ { 2 } } { y ^ { 6 } } \right)log4​(y6x2​)

Use the Quotient Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - ln⁡(e35)\ln \left( \frac { e ^ { 3 } } { 5 } \right)ln(5e3​)

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places - log⁡0.417\log _ { 0.4 } 17log0.4​17

Solve the equation by expressing each side as a power of the same base and then equating exponents. - 3x=273 x = 273x=27

Use the Product Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - log⁡6(7⋅3)\log _ { 6 } ( 7 \cdot 3 )log6​(7⋅3)

Use the Definition of a Logarithm to Solve Logarithmic Equations 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⁡6(x−1)=−3\log _ { 6 } ( x - 1 ) = - 3log6​(x−1)=−3

Solve the exponential equation. Express the solution set in terms of natural logarithms. - 39x=2.33 ^ { 9 x } = 2.339x=2.3

Graph the function. -Use the graph of log⁡5x\log _ { 5 } xlog5​x to obtain the graph of f(x)=12log⁡5xf ( x ) = \frac { 1 } { 2 } \log _ { 5 } xf(x)=21​log5​x .

Evaluate the expression without using a calculator. - log⁡4116\log _ { 4 } \frac { 1 } { 16 }log4​161​

Use the Definition of a Logarithm to Solve Logarithmic Equations 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+8=7\ln \sqrt { x + 8 } = 7lnx+8​=7

Graph the function. -Use the graph of f(x)=ln⁡xf ( x ) = \ln xf(x)=lnx to obtain the graph of g(x)=4−ln⁡xg ( x ) = 4 - \ln xg(x)=4−lnx .

Find the domain of the logarithmic function. - f(x)=ln⁡(6−x)f ( x ) = \ln ( 6 - x )f(x)=ln(6−x)

Graph the function. -Use the graph of f(x)=ln⁡xf ( x ) = \ln xf(x)=lnx to obtain the graph of g(x)=3ln⁡xg ( x ) = 3 \ln xg(x)=3lnx .