Question 1

Solve the equation.
-$$\left( \frac { 256 } { 81 } \right) ^ { x + 1 } = \left( \frac { 3 } { 4 } \right) ^ { x - 1 }$$

Accepted Answer

A)  $\left\{ - \frac { 6 } { 5 } \right\}$ 
B)  $\left\{ \frac { 3 } { 5 } \right\}$ 
C)  $\left\{ - \frac { 3 } { 5 } \right\}$ 
D)  $\left\{ - \frac { 3 } { 10 } \right\}$ 
A)  $\left\{ - \frac { 6 } { 5 } \right\}$ 
B)  $\left\{ \frac { 3 } { 5 } \right\}$ 
C)  $\left\{ - \frac { 3 } { 5 } \right\}$ 
D)  $\left\{ - \frac { 3 } { 10 } \right\}$

Question 2

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

Accepted Answer

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

Question 3

Find the future value.
-$59,878 invested for 3 years at 5% compounded semiannually 159)

Accepted Answer

A)  $69,316.27 
B)  $69,546.63 
C)  $69,440.12 
D)  $69,503.66 
A)  $69,316.27 
B)  $69,546.63 
C)  $69,440.12 
D)  $69,503.66

Question 4

Choose the one alternative that best completes the statement or answers the question.
-Suppose the consumption of electricity grows at $6.3 \%$ per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth.

Accepted Answer

A)  $17.44 \mathrm { yr }$ 
B)  $0.17 \mathrm { yr }$ 
C)  $47.62 \mathrm { yr }$ 
D)  $1.74 \mathrm { yr }$ 
A)  $17.44 \mathrm { yr }$ 
B)  $0.17 \mathrm { yr }$ 
C)  $47.62 \mathrm { yr }$ 
D)  $1.74 \mathrm { yr }$

Question 5

Solve the equation and express the solution in exact form.
-$\ln 3 x + \ln 9 x = \ln 28$

Accepted Answer

A)  $\{ 1 \}$ 
B)  $\left\{ \frac { \mathrm { e } ^ { 28 } } { 27 } \right\}$ 
C)  $\left\{ \left( \frac { 28 } { 27 } \right) ^ { 1 / 2 } \right\}$ 
D)  $\{ 0 \}$ 
A)  $\{ 1 \}$ 
B)  $\left\{ \frac { \mathrm { e } ^ { 28 } } { 27 } \right\}$ 
C)  $\left\{ \left( \frac { 28 } { 27 } \right) ^ { 1 / 2 } \right\}$ 
D)  $\{ 0 \}$

Question 6

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $$a \neq 1 \text { and } b \neq 1$$
-$\log _ { 3 } 8 - \log _ { 3 } z$

Accepted Answer

A)  $\log _ { 3 } ( 8 - z )$ 
B)  $\log _ { 6 } \left( \frac { 8 } { z } \right)$ 
C)  $\log _ { 3 } \left( \frac { 8 } { z } \right)$ 
D)  $\log _ { 3 } \left( \frac { z } { 8 } \right)$ 
A)  $\log _ { 3 } ( 8 - z )$ 
B)  $\log _ { 6 } \left( \frac { 8 } { z } \right)$ 
C)  $\log _ { 3 } \left( \frac { 8 } { z } \right)$ 
D)  $\log _ { 3 } \left( \frac { z } { 8 } \right)$

Question 7

Choose the one alternative that best completes the statement or answers the question.
-A population is increasing according to the exponential function defined by $y = 3 \mathrm { e } \cdot 04 \mathrm { x }$, where $\mathrm { y }$ is in millions and $\mathrm { x }$ is the number of years. Which of the following should be done in order to answer the question "How large will the population be in 4 years?"

Accepted Answer

A)  Evaluate $y = 3 \mathrm { e } ^ { - 04 ( 4 ) }$. 
B)  Solve $3 \mathrm { e } ^ { \cdot 04 \mathrm { x } } = 12$. 
C)  Evaluate $y = 3 \mathrm { e } ^ { - 04 ( 1 / 4 ) }$ 
D)  Solve $3 \mathrm { e } ^ { .04 \mathrm { x } } = 4$. 
A)  Evaluate $y = 3 \mathrm { e } ^ { - 04 ( 4 ) }$. 
B)  Solve $3 \mathrm { e } ^ { \cdot 04 \mathrm { x } } = 12$. 
C)  Evaluate $y = 3 \mathrm { e } ^ { - 04 ( 1 / 4 ) }$ 
D)  Solve $3 \mathrm { e } ^ { .04 \mathrm { x } } = 4$.

Question 8

Use properties of logarithms to evaluate the expression.
-If $f ( x ) = 5 ^ { x }$, find $f \left( \log _ { 5 } 7 \right)$

Accepted Answer

A)  7 
B)  $\log _ { 7 } x$ 
C)  5 
D)  35 
A)  7 
B)  $\log _ { 7 } x$ 
C)  5 
D)  35

Question 9

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

Question 10

The graph of a function f is given. Use the graph to find the indicated value.
-$$\mathrm { f } ^ { - 1 } ( 1 )$$

Accepted Answer

A)  2 
B)  4 
C)  3 
D)  $- 1$ 
A)  2 
B)  4 
C)  3 
D)  $- 1$

Question 11

Write in logarithmic form.
-$$32 ^ { 2 / 5 } = 4$$

Accepted Answer

A)  $\log _ { 32 } 4 = \frac { 2 } { 5 }$ 
B)  $\frac { \log _ { 5 } 4 } { \log _ { 2 } 32 } = 32$ 
C)  $\log _ { 2 } 32 = \frac { 2 } { 5 }$ 
D)  $\log _ { 4 } 32 = \frac { 2 } { 5 }$ 
A)  $\log _ { 32 } 4 = \frac { 2 } { 5 }$ 
B)  $\frac { \log _ { 5 } 4 } { \log _ { 2 } 32 } = 32$ 
C)  $\log _ { 2 } 32 = \frac { 2 } { 5 }$ 
D)  $\log _ { 4 } 32 = \frac { 2 } { 5 }$

Question 12

Solve the problem.
-The population of a particular city is increasing at a rate proportional to its size. It follows the function $\mathrm { P } ( \mathrm { t } ) = 1 + \mathrm { ke } ^ { 0.12 \mathrm { t } }$ where $\mathrm { k }$ is a constant and $\mathrm { t }$ is the time in years. If the current population is 18,000 , in how many years is the population expected to be 45,000 ? Round to the nearest year.

Accepted Answer

A)  $53 \mathrm { yr }$ 
B)  $3 \mathrm { yr }$ 
C)  $8 \mathrm { yr }$ 
D)  $5 \mathrm { yr }$ 
A)  $53 \mathrm { yr }$ 
B)  $3 \mathrm { yr }$ 
C)  $8 \mathrm { yr }$ 
D)  $5 \mathrm { yr }$

Question 13

Solve the problem.
-How long must $4700 be in a bank at 6% compounded annually to become $7067.06? (Round to the nearest year.)

Accepted Answer

A)  9 yr 
B)  6 yr 
C)  8 yr 
D)  7 yr 
A)  9 yr 
B)  6 yr 
C)  8 yr 
D)  7 yr

Question 14

Solve the equation.
-$$25 = b ^ { 2 / 3 }$$

Accepted Answer

A)  $\{ 125 \}$ 
B)  $\{ - 125,125 \}$ 
C)  $\{ 5 \}$ 
D)  $\{ - 5,5 \}$ 
A)  $\{ 125 \}$ 
B)  $\{ - 125,125 \}$ 
C)  $\{ 5 \}$ 
D)  $\{ - 5,5 \}$

Question 15

Solve the equation.
-$$5 ^ { - x } = \frac { 1 } { 625 }$$

Accepted Answer

A)  $\left\{ \frac { 1 } { 4 } \right\}$ 
B)  $\left\{ \frac { 1 } { 125 } \right\}$ 
C)  $\{ 4 \}$ 
D)  $\{ - 4 \}$ 
A)  $\left\{ \frac { 1 } { 4 } \right\}$ 
B)  $\left\{ \frac { 1 } { 125 } \right\}$ 
C)  $\{ 4 \}$ 
D)  $\{ - 4 \}$

Question 16

Provide an appropriate response.
-Give an equation of the form $f ( x ) = a ^ { x }$ to define the exponential function whose graph contains the point $\left( 4 , \frac { 1 } { 1296 } \right)$. Assume that $\mathrm { a } > 0$.

Accepted Answer

A)  $f ( x ) = 6 ^ { x }$ 
B)  $f ( x ) = \left( \frac { 1 } { 4 } \right) ^ { x }$ 
C)  $f ( x ) = 4 ^ { x }$ 
D)  $f ( x ) = \left( \frac { 1 } { 6 } \right) ^ { x }$ 
A)  $f ( x ) = 6 ^ { x }$ 
B)  $f ( x ) = \left( \frac { 1 } { 4 } \right) ^ { x }$ 
C)  $f ( x ) = 4 ^ { x }$ 
D)  $f ( x ) = \left( \frac { 1 } { 6 } \right) ^ { x }$

Question 17

Write the word or phrase that best completes each statement or answers the question.
-$$\text { Explain why } \log _ { 2 } 13 \text { is between } 3 \text { and } 4$$

Accepted Answer

@#LAT-DLM& and @#LAT-DLM&. Since @#LAT-DLM&, then

Question 18

Use the definition of inverses to determine whether f and g are inverses.
-$$f ( x ) = 9 x - 2 , \quad g ( x ) = \frac { x + 9 } { 2 }$$

Accepted Answer

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

Question 19

The graph of a function f is given. Use the graph to find the indicated value.
-$f^{-1}(2)$

Accepted Answer

A)  2 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  6 
A)  2 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 3 } { 2 }$ 
D)  6

Question 20

Solve for the indicated variable.
-$\mathrm { v } = \mathrm { p } - \mathrm { k } \ln \mathrm { t }$, for $\mathrm { t }$

Accepted Answer

A)  $t = e ( p - v ) / k$ 
B)  $t = \frac { p - v } { k }$ 
C)  $t = \ln \frac { p - v } { k }$ 
D)  $t = e ^ { ( v - p ) / k }$ 
A)  $t = e ( p - v ) / k$ 
B)  $t = \frac { p - v } { k }$ 
C)  $t = \ln \frac { p - v } { k }$ 
D)  $t = e ^ { ( v - p ) / k }$

Solve the equation. - $\left( \frac { 256 } { 81 } \right) ^ { x + 1 } = \left( \frac { 3 } { 4 } \right) ^ { x - 1 }$

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

Find the future value. -$59,878 invested for 3 years at 5% compounded semiannually 159)

Choose the one alternative that best completes the statement or answers the question. -Suppose the consumption of electricity grows at $6.3 \%$ per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth.

Solve the equation and express the solution in exact form. - $\ln 3 x + \ln 9 x = \ln 28$

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ - $\log _ { 3 } 8 - \log _ { 3 } z$

Use properties of logarithms to evaluate the expression. -If $f ( x ) = 5 ^ { x }$ , find $f \left( \log _ { 5 } 7 \right)$

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

The graph of a function f is given. Use the graph to find the indicated value. - $\mathrm { f } ^ { - 1 } ( 1 )$ $The graph of a function f is given. Use the graph to find the indicated value. - \mathrm { f } ^ { - 1 } ( 1 )$

Write in logarithmic form. - $32 ^ { 2 / 5 } = 4$

Solve the problem. -How long must $4700 be in a bank at 6% compounded annually to become $7067.06? (Round to the nearest year.)

Solve the equation. - $25 = b ^ { 2 / 3 }$

Solve the equation. - $5 ^ { - x } = \frac { 1 } { 625 }$

Provide an appropriate response. -Give an equation of the form $f ( x ) = a ^ { x }$ to define the exponential function whose graph contains the point $\left( 4 , \frac { 1 } { 1296 } \right)$ . Assume that $\mathrm { a } > 0$ .

Write the word or phrase that best completes each statement or answers the question. - $\text { Explain why } \log _ { 2 } 13 \text { is between } 3 \text { and } 4$

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = 9 x - 2 , \quad g ( x ) = \frac { x + 9 } { 2 }$

The graph of a function f is given. Use the graph to find the indicated value. - $f^{-1}(2)$ $The graph of a function f is given. Use the graph to find the indicated value. - f^{-1}(2)$

Solve for the indicated variable. - $\mathrm { v } = \mathrm { p } - \mathrm { k } \ln \mathrm { t }$ , for $\mathrm { t }$

Review of Basic Concepts

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

Filters

Exam 5: Inverse, Exponential, and Logarithmic Functions

Solve the equation. - (25681)x+1=(34)x−1\left( \frac { 256 } { 81 } \right) ^ { x + 1 } = \left( \frac { 3 } { 4 } \right) ^ { x - 1 }(81256​)x+1=(43​)x−1

Find the domain and range of the inverse of the given function. - f(x)=3x−4f ( x ) = \frac { 3 } { x - 4 }f(x)=x−43​

Find the future value. -$59,878 invested for 3 years at 5% compounded semiannually 159)

Choose the one alternative that best completes the statement or answers the question. -Suppose the consumption of electricity grows at 6.3%6.3 \%6.3% per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth.

Solve the equation and express the solution in exact form. - ln⁡3x+ln⁡9x=ln⁡28\ln 3 x + \ln 9 x = \ln 28ln3x+ln9x=ln28

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with a≠1 and b≠1a \neq 1 \text { and } b \neq 1a=1 and b=1 - log⁡38−log⁡3z\log _ { 3 } 8 - \log _ { 3 } zlog3​8−log3​z

Use properties of logarithms to evaluate the expression. -If f(x)=5xf ( x ) = 5 ^ { x }f(x)=5x , find f(log⁡57)f \left( \log _ { 5 } 7 \right)f(log5​7)

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)

The graph of a function f is given. Use the graph to find the indicated value. - f−1(1)\mathrm { f } ^ { - 1 } ( 1 )f−1(1)

Write in logarithmic form. - 322/5=432 ^ { 2 / 5 } = 4322/5=4