Question 1

Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$5 \mathrm { e } ^ { \mathrm { x } } = 23$

Accepted Answer

A)  $1.53$ 
B)  $0.66$ 
C)  $- 1.53$ 
D)  $- 0.66$ 
A)  $1.53$ 
B)  $0.66$ 
C)  $- 1.53$ 
D)  $- 0.66$

Question 2

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log \left( \frac { x } { 100 } \right)$

Accepted Answer

A)  $\log x - 2$ 
B)  $\log x + 2$ 
C)  $- 20 x$ 
D)  $100 x$ 
A)  $\log x - 2$ 
B)  $\log x + 2$ 
C)  $- 20 x$ 
D)  $100 x$

Question 3

Solve the equation by expressing each side as a power of the same base and then equating exponents.
-$$2 ( 3 x + 5 ) = \frac { 1 } { 16 }$$

Accepted Answer

A)  $\{ - 3 \}$ 
B)  $\left\{ \frac { 1 } { 8 } \right\}$ 
C)  $\{ 8 \}$ 
D)  $\{ 3 \}$ 
A)  $\{ - 3 \}$ 
B)  $\left\{ \frac { 1 } { 8 } \right\}$ 
C)  $\{ 8 \}$ 
D)  $\{ 3 \}$

Question 4

Find the domain of the logarithmic function.
-$$f ( x ) = \log _ { 4 } ( x - 8 )$$

Accepted Answer

A)  $( 8 , \infty )$ 
B)  $( - \infty , 8 )$ or $( 8 , \infty )$ 
C)  $( - 8 , \infty )$ 
D)  $( - \infty , 0 )$ or $( 0 , \infty )$ 
A)  $( 8 , \infty )$ 
B)  $( - \infty , 8 )$ or $( 8 , \infty )$ 
C)  $( - 8 , \infty )$ 
D)  $( - \infty , 0 )$ or $( 0 , \infty )$

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 $\mathrm { A } = 2000 \mathrm { e } ^ { 0.048 \mathrm { t } }$. By what percentage is the account increasing each year?

Accepted Answer

A)  $4.8 \%$ 
B)  $5.1 \%$ 
C)  $5.3 \%$ 
D)  $5.4 \%$ 
A)  $4.8 \%$ 
B)  $5.1 \%$ 
C)  $5.3 \%$ 
D)  $5.4 \%$

Question 6

Evaluate or simplify the expression without using a calculator.
-$\log 0.001$

Accepted Answer

A)  $- 3$ 
B)  3 
C)  $- \frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 3 }$ 
A)  $- 3$ 
B)  3 
C)  $- \frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 3 }$

Question 7

Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$$2 ^ { 7 x } = 4.6$$

Accepted Answer

A)  $0.31$ 
B)  $0.39$ 
C)  $15.41$ 
D)  $12.91$ 
A)  $0.31$ 
B)  $0.39$ 
C)  $15.41$ 
D)  $12.91$

Question 8

Write the equation in its equivalent logarithmic form.
-$$2 ^ { 3 } = x$$

Accepted Answer

A)  $\log _ { 2 } x = 3$ 
B)  $\log _ { x } 2 = 3$ 
C)  $\log _ { 3 } x = 2$ 
D)  $\log _ { 2 } 3 = x$ 
A)  $\log _ { 2 } x = 3$ 
B)  $\log _ { x } 2 = 3$ 
C)  $\log _ { 3 } x = 2$ 
D)  $\log _ { 2 } 3 = x$

Question 9

Evaluate or simplify the expression without using a calculator.
-The $\mathrm { pH }$ of a solution ranges from 0 to 14 . An acid has a pH less than 7 . Pure water is neutral and has a pH of 7. The $\mathrm { pH }$ of a solution is given by $\mathrm { pH } = - \log x$ where $x$ represents the concentration of the hydrogen ions in the solution in moles per liter. Find the $\mathrm { pH }$ if the hydrogen ion concentration is $6.4 \times 10 ^ { - 3 }$.

Accepted Answer

A)  $2.19$ 
B)  $3.81$ 
C)  $3.19$ 
D)  $2.81$ 
A)  $2.19$ 
B)  $3.81$ 
C)  $3.19$ 
D)  $2.81$

Question 10

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.
-$$\log _ { 4 } ( x + 2 ) + \log _ { 4 } ( x - 4 ) = 2$$

Accepted Answer

A)  $\{ 6 \}$ 
B)  $\{ 6 , - 4 \}$ 
C)  $\{ - 4 \}$ 
D)  $\{ 7 \}$ 
A)  $\{ 6 \}$ 
B)  $\{ 6 , - 4 \}$ 
C)  $\{ - 4 \}$ 
D)  $\{ 7 \}$

Question 11

Solve the problem.
-The logistic growth function $\mathrm { f } ( \mathrm { t } ) = \frac { 320 } { 1 + 4.3 \mathrm { e } ^ { - 0.21 \mathrm { t } } }$ 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)  60 butterflies 
B)  320 butterflies 
C)  4 butterflies 
D)  2 butterflies 
A)  60 butterflies 
B)  320 butterflies 
C)  4 butterflies 
D)  2 butterflies

Question 12

Solve.
-The half-life of silicon-32 is 710 years. If 80 grams is present now, how much will be present in 200 years? (Round your answer to three decimal places.)

Accepted Answer

A)  $65.81$ 
B)  $78.453$ 
C)  $11.353$ 
D)  0 
A)  $65.81$ 
B)  $78.453$ 
C)  $11.353$ 
D)  0

Question 13

Solve the problem.
-The function $\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.00693 \mathrm { x } }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. If 800 pounds of the material are placed in the vault, how much time will need to pass for only 492 pounds to remain?

Accepted Answer

A)  70 years 
B)  75 years 
C)  80 years 
D)  140 years 
A)  70 years 
B)  75 years 
C)  80 years 
D)  140 years

Question 14

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.
-$$y = 700 ( 1.9 ) ^ { x }$$

Accepted Answer

A)  $y = 700 e ^ { x } \ln 1.9 , y = 700 e ^ { 0.642 x }$ 
B)  $y = 1.9 e ^ { x } \ln 700 , y = 1.9 e ^ { 6.551 x }$ 
C)  $\mathrm { y } = 700 \mathrm { e } ^ { 1.9 x } , \mathrm { y } = 7002.718 ^ { 0.642 \mathrm { x } }$ 
D)  $y = ( \ln 700 ) \mathrm { e } ^ { \mathrm { x } \ln 1.9 } , \mathrm { y } = 6.551 \mathrm { e } ^ { 0.642 \mathrm { x } }$ 
A)  $y = 700 e ^ { x } \ln 1.9 , y = 700 e ^ { 0.642 x }$ 
B)  $y = 1.9 e ^ { x } \ln 700 , y = 1.9 e ^ { 6.551 x }$ 
C)  $\mathrm { y } = 700 \mathrm { e } ^ { 1.9 x } , \mathrm { y } = 7002.718 ^ { 0.642 \mathrm { x } }$ 
D)  $y = ( \ln 700 ) \mathrm { e } ^ { \mathrm { x } \ln 1.9 } , \mathrm { y } = 6.551 \mathrm { e } ^ { 0.642 \mathrm { x } }$

Question 15

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

Question 16

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log ( 10,000 x )$

Accepted Answer

A)  $4 + \log x$ 
B)  $40 + \log x$ 
C)  $4 x$ 
D)  $4 \log x$ 
A)  $4 + \log x$ 
B)  $40 + \log x$ 
C)  $4 x$ 
D)  $4 \log x$

Question 17

Graph the function
-Use the graph of $f ( x ) = e ^ { x }$ to obtain the graph of $g ( x ) = e ^ { 4 x }$.

Accepted Answer

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

Question 18

Graph the function.
-Use the graph of $f ( x ) = \log x$ to obtain the graph of $g ( x ) = \log ( x - 1 )$.

Accepted Answer

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

Question 19

Solve the problem.
-The logistic growth function $f ( t ) = \frac { 600 } { 1 + 6.5 e ^ { - 0.18 t } }$ describes the population of a species of butterflies t months after they are introduced to a non-threatening habitat. How many butterflies are expected in the habitat after 12 months?

Accepted Answer

A)  343 butterflies 
B)  960 butterflies 
C)  601 butterflies 
D)  7200 butterflies 
A)  343 butterflies 
B)  960 butterflies 
C)  601 butterflies 
D)  7200 butterflies

Question 20

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.
-$$\log _ { 2 } ( 3 x + 5 ) = \log _ { 2 } ( 3 x + 8 )$$

Accepted Answer

A)  $\{ 3 \}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { 13 } { 3 } \right\}$ 
D)  $\varnothing$ 
A)  $\{ 3 \}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { 13 } { 3 } \right\}$ 
D)  $\varnothing$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $5 \mathrm { e } ^ { \mathrm { x } } = 23$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log \left( \frac { x } { 100 } \right)$

Solve the equation by expressing each side as a power of the same base and then equating exponents. - $2 ( 3 x + 5 ) = \frac { 1 } { 16 }$

Find the domain of the logarithmic function. - $f ( x ) = \log _ { 4 } ( x - 8 )$

Evaluate or simplify the expression without using a calculator. - $\log 0.001$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $2 ^ { 7 x } = 4.6$

Write the equation in its equivalent logarithmic form. - $2 ^ { 3 } = x$

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. - $\log _ { 4 } ( x + 2 ) + \log _ { 4 } ( x - 4 ) = 2$

Solve. -The half-life of silicon-32 is 710 years. If 80 grams is present now, how much will be present in 200 years? (Round your answer to three decimal places.)

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. - $y = 700 ( 1.9 ) ^ { x }$

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

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log ( 10,000 x )$

Graph the function -Use the graph of $f ( x ) = e ^ { x }$ to obtain the graph of $g ( x ) = e ^ { 4 x }$ . $Graph the function -Use the graph of f ( x ) = e ^ { x } to obtain the graph of g ( x ) = e ^ { 4 x } .$

Graph the function. -Use the graph of $f ( x ) = \log x$ to obtain the graph of $g ( x ) = \log ( x - 1 )$ . $Graph the function. -Use the graph of f ( x ) = \log x to obtain the graph of g ( x ) = \log ( x - 1 ) .$

Solve the problem. -The logistic growth function $f ( t ) = \frac { 600 } { 1 + 6.5 e ^ { - 0.18 t } }$ describes the population of a species of butterflies t months after they are introduced to a non-threatening habitat. How many butterflies are expected in the habitat after 12 months?

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. - $\log _ { 2 } ( 3 x + 5 ) = \log _ { 2 } ( 3 x + 8 )$

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. - 5ex=235 \mathrm { e } ^ { \mathrm { x } } = 235ex=23

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - log⁡(x100)\log \left( \frac { x } { 100 } \right)log(100x​)

Solve the equation by expressing each side as a power of the same base and then equating exponents. - 2(3x+5)=1162 ( 3 x + 5 ) = \frac { 1 } { 16 }2(3x+5)=161​

Find the domain of the logarithmic function. - f(x)=log⁡4(x−8)f ( x ) = \log _ { 4 } ( x - 8 )f(x)=log4​(x−8)

Evaluate or simplify the expression without using a calculator. - log⁡0.001\log 0.001log0.001

Solve the exponential equation. Express the solution set in terms of natural logarithms. - 27x=4.62 ^ { 7 x } = 4.627x=4.6

Write the equation in its equivalent logarithmic form. - 23=x2 ^ { 3 } = x23=x

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⁡4(x+2)+log⁡4(x−4)=2\log _ { 4 } ( x + 2 ) + \log _ { 4 } ( x - 4 ) = 2log4​(x+2)+log4​(x−4)=2