Question 1

Use the definition of inverses to determine whether f and g are inverses.
-$$f ( x ) = x ^ { 3 } - 3 , \quad g ( x ) = \sqrt [ 3 ] { x + 3 }$$

Accepted Answer

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

Question 2

If the function is one-to-one, find its inverse. If not, write "not one-to-one."
-$$f ( x ) = ( x + 6 ) ^ { 2 }$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \frac { 1 } { \sqrt { x - 6 } }$ 
B)  not a one-to-one 
C)  $f ^ { - 1 } ( x ) = \sqrt { x - 6 }$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt { \mathrm { x } } - 6$ 
A)  $f ^ { - 1 } ( x ) = \frac { 1 } { \sqrt { x - 6 } }$ 
B)  not a one-to-one 
C)  $f ^ { - 1 } ( x ) = \sqrt { x - 6 }$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt { \mathrm { x } } - 6$

Question 3

Use the change of base rule to find the logarithm to four decimal places.
-$$\log _ { 4.2 } 3.2$$

Accepted Answer

A)  0.5051 
B)  0.7619 
C)  1.2338 
D)  0.8105 
A)  0.5051 
B)  0.7619 
C)  1.2338 
D)  0.8105

Question 4

Solve the problem.
-Which of the following is the same as $3 \log ( 4 x )$ for $x > 0$ ?

Accepted Answer

A)  $\log 12 + \log x$ 
B)  $\log \left( 64 x ^ { 3 } \right)$ 
C)  $\log 64 \cdot \log x ^ { 3 }$ 
D)  $\log ( 12 x )$ 
A)  $\log 12 + \log x$ 
B)  $\log \left( 64 x ^ { 3 } \right)$ 
C)  $\log 64 \cdot \log x ^ { 3 }$ 
D)  $\log ( 12 x )$

Question 5

Solve the equation and express the solution in exact form.
-$\log 3 x = \log 2 + \log ( x + 4 )$

Accepted Answer

A)  $\{ 8 \}$ 
B)  $\left\{ \frac { 8 } { 5 } \right\}$ 
C)  $\{ 3 \}$ 
D)  $\{ 4 \}$ 
A)  $\{ 8 \}$ 
B)  $\left\{ \frac { 8 } { 5 } \right\}$ 
C)  $\{ 3 \}$ 
D)  $\{ 4 \}$

Question 6

Use properties of logarithms to evaluate the expression.
-$\log _ { 10 } ( 0.01 ) ^ { 3 }$

Accepted Answer

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

Question 7

Given $$\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$$ find the logarithm without using a calculator.
-$$\log _ { 10 } \frac { 27 } { 4 }$$

Accepted Answer

A)  $0.0512$ 
B)  $1.8572$ 
C)  $0.2549$ 
D)  $0.8293$ 
A)  $0.0512$ 
B)  $1.8572$ 
C)  $0.2549$ 
D)  $0.8293$

Question 8

If the function is one-to-one, find its inverse. If not, write "not one-to-one."
-{(-3, -4), (-2, -4), (-1, 3), (0, 8)}

Accepted Answer

A)  {(-4, -3), (8, 3), (-1, -2)} 
B)  {(-4, -3), (-2, 3), (-1, 8)} 
C)  {(-3, 8), (-3, 3), (-1, -2)} 
D)  not one-to-one 
A)  {(-4, -3), (8, 3), (-1, -2)} 
B)  {(-4, -3), (-2, 3), (-1, 8)} 
C)  {(-3, 8), (-3, 3), (-1, -2)} 
D)  not one-to-one

Question 9

Solve the problem.
-The decibel level $\mathrm { D }$ of a sound is related to its intensity $\mathrm { I }$ by $\mathrm { D } = 10 \log \left( \frac { \mathrm { I } } { \mathrm { I } _ { 0 } } \right)$. If $\mathrm { I } _ { 0 }$ is $10 ^ { - 12 }$, then what is the intensity of a noise measured at 78 decibels? Express your answer in scientific notation, rounding to three significant digits, if necessary.

Accepted Answer

A)  $7.8 \times 10 ^ { - 10 } \mathrm { Watt } / \mathrm { m } ^ { 2 }$ 
B)  $2.44 \times 10 ^ { 15 }$ watt $/ \mathrm { m } ^ { 2 }$ 
C)  $6.31 \times 10 ^ { - 4 }$ watt $/ \mathrm { m } ^ { 2 }$ 
D)  $6.31 \times 10 ^ { - 5 }$ watt $/ \mathrm { m } ^ { 2 }$ 
A)  $7.8 \times 10 ^ { - 10 } \mathrm { Watt } / \mathrm { m } ^ { 2 }$ 
B)  $2.44 \times 10 ^ { 15 }$ watt $/ \mathrm { m } ^ { 2 }$ 
C)  $6.31 \times 10 ^ { - 4 }$ watt $/ \mathrm { m } ^ { 2 }$ 
D)  $6.31 \times 10 ^ { - 5 }$ watt $/ \mathrm { m } ^ { 2 }$

Question 10

Find the value. Give an approximation to four decimal places.
-$\ln 0.000814$

Accepted Answer

A)  $3.0894$ 
B)  $7.1136$ 
C)  $- 3.0894$ 
D)  $- 7.1136$ 
A)  $3.0894$ 
B)  $7.1136$ 
C)  $- 3.0894$ 
D)  $- 7.1136$

Question 11

Solve the equation. Round to the nearest thousandth.
-$$9 \mathrm { e } ^ { ( 3 x + 9 ) } = 5$$

Accepted Answer

A)  $\{ - 4.726 \}$ 
B)  $\{ - 4.334 \}$ 
C)  $\{ - 1.666 \}$ 
D)  $\{ - 3.196 \}$ 
A)  $\{ - 4.726 \}$ 
B)  $\{ - 4.334 \}$ 
C)  $\{ - 1.666 \}$ 
D)  $\{ - 3.196 \}$

Question 12

Choose the one alternative that best completes the statement or answers the question.
-The hydrogen potential, $\mathrm { pH }$, of a substance is defined by $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$, where $\left[ \mathrm { H } ^ { + } \right]$is measured in moles per liter. Find the pH of a sample of lake water whose $\left[ \mathrm { H } ^ { + } \right]$is $3.05 \times 10 ^ { - 9 }$ moles per liter. (Round to the nearest tenth.)

Accepted Answer

A)  $6.4$ 
B)  $10.1$ 
C)  $8.5$ 
D)  $7.3$ 
A)  $6.4$ 
B)  $10.1$ 
C)  $8.5$ 
D)  $7.3$

Question 13

rite the word or phrase that best completes each statement or answers the question.
-Explain how to solv $$e ^ { 2 x } + 8 e ^ { x } - 20 = 0 \text { analytically. }$$

Accepted Answer

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

Question 14

Solve the equation and express the solution in exact form.
-$$\log _ { 4 } \left( \log _ { 4 } x \right) = 1$$

Accepted Answer

A)  $\{ 256 \}$ 
B)  $\{ 8 \}$ 
C)  $\{ 16 \}$ 
D)  $\{ 4 \}$ 
A)  $\{ 256 \}$ 
B)  $\{ 8 \}$ 
C)  $\{ 16 \}$ 
D)  $\{ 4 \}$

Question 15

Solve the problem.
-The height in meters of women of a certain tribe is approximated by $h = 0.52 + 2 \log ( t / 3 )$ where $t$ is the woman's age in years and $1 \leq t \leq 20$. Estimate the height (to the nearest hundredth) of a woman of the tribe 4 years of age.

Accepted Answer

A)  $0.96 \mathrm {~m}$ 
B)  $1.12 \mathrm {~m}$ 
C)  $0.77 \mathrm {~m}$ 
D)  $0.52 \mathrm {~m}$ 
A)  $0.96 \mathrm {~m}$ 
B)  $1.12 \mathrm {~m}$ 
C)  $0.77 \mathrm {~m}$ 
D)  $0.52 \mathrm {~m}$

Question 16

Solve the equation.
-$$e ^ { x - 5 } = \left( \frac { 1 } { e ^ { 6 } } \right) ^ { x + 4 }$$

Accepted Answer

A)  $\left\{ - \frac { 29 } { 5 } \right\}$ 
B)  $\left\{ - \frac { 19 } { 7 } \right\}$ 
C)  $\left\{ \frac { 9 } { 7 } \right\}$ 
D)  $\left\{ - \frac { 9 } { 5 } \right\}$ 
A)  $\left\{ - \frac { 29 } { 5 } \right\}$ 
B)  $\left\{ - \frac { 19 } { 7 } \right\}$ 
C)  $\left\{ \frac { 9 } { 7 } \right\}$ 
D)  $\left\{ - \frac { 9 } { 5 } \right\}$

Question 17

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent
positive real numbers.
-$$\log b \left( \frac { m ^ { 2 } p ^ { 6 } } { n ^ { 4 } b ^ { 8 } } \right)$$

Accepted Answer

A)  $m ^ { 2 } p ^ { 6 } - n ^ { 4 } b ^ { 8 }$ 
B)  $\log _ { b } m ^ { 2 } + \log _ { b } p ^ { 6 } + \log _ { b } n ^ { 4 } - \log _ { b } b ^ { 8 }$ 
C)  $2 \log _ { b } m + 6 \log _ { b } p - 4 \log _ { b } n - 8$ 
D)  $2 \log _ { b } m + 6 \log _ { b } p - 4 \log _ { b } n + 8$ 
A)  $m ^ { 2 } p ^ { 6 } - n ^ { 4 } b ^ { 8 }$ 
B)  $\log _ { b } m ^ { 2 } + \log _ { b } p ^ { 6 } + \log _ { b } n ^ { 4 } - \log _ { b } b ^ { 8 }$ 
C)  $2 \log _ { b } m + 6 \log _ { b } p - 4 \log _ { b } n - 8$ 
D)  $2 \log _ { b } m + 6 \log _ { b } p - 4 \log _ { b } n + 8$

Question 18

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $$a \neq 1 \text { and } b \neq 1$$
-$$\frac { 1 } { 3 } \log _ { 3 } \left( x ^ { 6 } \right) + \frac { 1 } { 6 } \log _ { 3 } \left( x ^ { 6 } \right) - \frac { 1 } { 9 } \log _ { 3 } x$$

Accepted Answer

A)  $\log _ { 3 } \left( x ^ { 7 } \right)$ 
B)  $\frac { 7 } { 9 } \log _ { 3 } \left( x ^ { 12 } \right)$ 
C)  $\log _ { 3 } \left( x ^ { 26 / 9 } \right)$ 
D)  $\log _ { 3 } \left( x ^ { 9 / 2 } \right)$ 
A)  $\log _ { 3 } \left( x ^ { 7 } \right)$ 
B)  $\frac { 7 } { 9 } \log _ { 3 } \left( x ^ { 12 } \right)$ 
C)  $\log _ { 3 } \left( x ^ { 26 / 9 } \right)$ 
D)  $\log _ { 3 } \left( x ^ { 9 / 2 } \right)$

Question 19

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

Accepted Answer

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

Question 20

Find the value. Give an approximation to four decimal places.
-$\ln 797 + \ln 27$

Accepted Answer

A)  $4.3343$ 
B)  $4.1530$ 
C)  $22.0190$ 
D)  $9.9767$ 
A)  $4.3343$ 
B)  $4.1530$ 
C)  $22.0190$ 
D)  $9.9767$

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = x ^ { 3 } - 3 , \quad g ( x ) = \sqrt [ 3 ] { x + 3 }$

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $f ( x ) = ( x + 6 ) ^ { 2 }$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 4.2 } 3.2$

Solve the problem. -Which of the following is the same as $3 \log ( 4 x )$ for $x > 0$ ?

Solve the equation and express the solution in exact form. - $\log 3 x = \log 2 + \log ( x + 4 )$

Use properties of logarithms to evaluate the expression. - $\log _ { 10 } ( 0.01 ) ^ { 3 }$

Given $\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$ find the logarithm without using a calculator. - $\log _ { 10 } \frac { 27 } { 4 }$

If the function is one-to-one, find its inverse. If not, write "not one-to-one." -{(-3, -4), (-2, -4), (-1, 3), (0, 8)}

Find the value. Give an approximation to four decimal places. - $\ln 0.000814$

Solve the equation. Round to the nearest thousandth. - $9 \mathrm { e } ^ { ( 3 x + 9 ) } = 5$

rite the word or phrase that best completes each statement or answers the question. -Explain how to solv $e ^ { 2 x } + 8 e ^ { x } - 20 = 0 \text { analytically. }$

Solve the equation and express the solution in exact form. - $\log _ { 4 } \left( \log _ { 4 } x \right) = 1$

Solve the problem. -The height in meters of women of a certain tribe is approximated by $h = 0.52 + 2 \log ( t / 3 )$ where $t$ is the woman's age in years and $1 \leq t \leq 20$ . Estimate the height (to the nearest hundredth) of a woman of the tribe 4 years of age.

Solve the equation. - $e ^ { x - 5 } = \left( \frac { 1 } { e ^ { 6 } } \right) ^ { x + 4 }$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log b \left( \frac { m ^ { 2 } p ^ { 6 } } { n ^ { 4 } b ^ { 8 } } \right)$

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

Find the value. Give an approximation to four decimal places. - $\ln 797 + \ln 27$

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

Use the definition of inverses to determine whether f and g are inverses. - f(x)=x3−3,g(x)=x+33f ( x ) = x ^ { 3 } - 3 , \quad g ( x ) = \sqrt [ 3 ] { x + 3 }f(x)=x3−3,g(x)=3x+3​

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - f(x)=(x+6)2f ( x ) = ( x + 6 ) ^ { 2 }f(x)=(x+6)2

Use the change of base rule to find the logarithm to four decimal places. - log⁡4.23.2\log _ { 4.2 } 3.2log4.2​3.2

Solve the problem. -Which of the following is the same as 3log⁡(4x)3 \log ( 4 x )3log(4x) for x>0x > 0x>0 ?

Solve the equation and express the solution in exact form. - log⁡3x=log⁡2+log⁡(x+4)\log 3 x = \log 2 + \log ( x + 4 )log3x=log2+log(x+4)

Use properties of logarithms to evaluate the expression. - log⁡10(0.01)3\log _ { 10 } ( 0.01 ) ^ { 3 }log10​(0.01)3

Given log⁡102≈0.3010 and log⁡103≈0.4771\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771log10​2≈0.3010 and log10​3≈0.4771 find the logarithm without using a calculator. - log⁡10274\log _ { 10 } \frac { 27 } { 4 }log10​427​