Question 1

Determine the function value.
-Suppose $f ( x ) = \log _ { a } x$ and $f ( 5 ) = 2$. Find $f \left( \frac { 1 } { 25 } \right)$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  4 
C)  $- 2$ 
D)  $- 4$ 
A)  $\frac { 1 } { 2 }$ 
B)  4 
C)  $- 2$ 
D)  $- 4$

Question 2

Find the present value of the future value.
-The growth in the mouse population at a certain county dump can be modeled by the exponential function $$\mathrm { A } ( \mathrm { t } ) = 554 \mathrm { e } ^ { 0.015 \mathrm { t } }$$ , where t is the number of months since the population was first recorded.
Estimate the population after 39 months.

Accepted Answer

A)  1009 
B)  994 
C)  562 
D)  497 
A)  1009 
B)  994 
C)  562 
D)  497

Question 3

Write an equivalent expression in exponential form.
-$$x = 3 ^ { \log _ { 3 } 14 }$$

Accepted Answer

A)  $\{ 42 \}$ 
B)  $\{ 14 \}$ 
C)  $\{ 3 \}$ 
D)  $\{ - 14 \}$ 
A)  $\{ 42 \}$ 
B)  $\{ 14 \}$ 
C)  $\{ 3 \}$ 
D)  $\{ - 14 \}$

Question 4

Solve the equation. Round to the nearest thousandth.
-$$53 x = 6 ( x + 1 )$$

Accepted Answer

A)  $\{ 0.590 \}$ 
B)  $\{ 1.113 \}$ 
C)  $\{ - 9.827 \}$ 
D)  $\{ 2.113 \}$ 
A)  $\{ 0.590 \}$ 
B)  $\{ 1.113 \}$ 
C)  $\{ - 9.827 \}$ 
D)  $\{ 2.113 \}$

Question 5

Find the present value of the future value.
-The number of reports of a certain virus has increased exponentially since year 0. The number of cases can be approximated using the function $$r ( t ) = 101 e ^ { 0.005 t }$$ where t is the number of years since
Year 0. Estimate the number of cases in year 40.

Accepted Answer

A)  247 cases 
B)  117 cases 
C)  123 cases 
D)  93 cases 
A)  247 cases 
B)  117 cases 
C)  123 cases 
D)  93 cases

Question 6

Solve the problem.
-The population growth of an animal species is described by $\mathrm { F } ( \mathrm { t } ) = 400 \log ( 2 \mathrm { t } + 3 )$ where $\mathrm { t }$ is measured in months. Find the population of this species in an area 6 months after the species is introduced.

Accepted Answer

A)  470 
B)  704 
C)  74 
D)  240 
A)  470 
B)  704 
C)  74 
D)  240

Question 7

rite the word or phrase that best completes each statement or answers the question.
-Write the algebraic equation which can be used to find the exact solution of $\log _ { 2 } 2 ( x + 4 ) =$
$$\frac { 40 } { x - 10 }$$

Accepted Answer

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

Question 8

Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] , \text {as }$ needed.
-Find $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$if the $\mathrm { pH } = 11$.

Accepted Answer

A)  $3.0 \times 10 ^ { - 11 }$ 
B)  $2.0 \times 1011$ 
C)  $1.0 \times 1011$ 
D)  $1.0 \times 10 ^ { - 11 }$ 
A)  $3.0 \times 10 ^ { - 11 }$ 
B)  $2.0 \times 1011$ 
C)  $1.0 \times 1011$ 
D)  $1.0 \times 10 ^ { - 11 }$

Question 9

Use properties of logarithms to evaluate the expression.
-If $\mathrm { f } ( \mathrm { x } ) = \log 7 x$, find $f \left( 7 ^ { 8 } \right)$

Accepted Answer

A)  8 
B)  $\log 8 x$ 
C)  7 
D)  56 
A)  8 
B)  $\log 8 x$ 
C)  7 
D)  56

Question 10

rite the word or phrase that best completes each statement or answers the question.
-Without solving the inequality, what are the restrictions on $x$ if $\log _ { 3 } ( x - 4 ) - \log _ { 3 } ( x + 1 ) <$ 3? Explain.

Accepted Answer

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

Question 11

Find the value. Give an approximation to four decimal places.
-$\ln 295$

Accepted Answer

A)  $108.8561$ 
B)  $2.4698$ 
C)  $0.1753$ 
D)  $5.6870$ 
A)  $108.8561$ 
B)  $2.4698$ 
C)  $0.1753$ 
D)  $5.6870$

Question 12

Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] , \text {as }$ needed.
-Find the $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$if the $\mathrm { pH } = 2.3$.

Accepted Answer

A)  $2.7 \times 10 ^ { 1 }$ 
B)  $5.0 \times 10 ^ { - 3 }$ 
C)  $2.0 \times 10 ^ { 2 }$ 
D)  $3.6 \times 10 ^ { - 1 }$ 
A)  $2.7 \times 10 ^ { 1 }$ 
B)  $5.0 \times 10 ^ { - 3 }$ 
C)  $2.0 \times 10 ^ { 2 }$ 
D)  $3.6 \times 10 ^ { - 1 }$

Question 13

Graph the exponential function using transformations where appropriate.
-$f(x)=-5^{x}$

Accepted Answer

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

Question 14

Graph the function. Give the domain and range.
-$$f ( x ) = \log _ { 1 / 2 } ( 2 - x )$$

Accepted Answer

A)  domain: $( - 2 , \infty )$; range: $( \infty , \infty )$
  
B)  domain: $( - 2 , \infty )$; range: $( \infty , \infty )$
  
C)  domain: $( \infty , 2 )$; range: $( \infty , \infty )$
  
D)  domain: $( \infty , 2 )$; range: $( \infty , \infty )$
  
A)  domain: $( - 2 , \infty )$; range: $( \infty , \infty )$
  
B)  domain: $( - 2 , \infty )$; range: $( \infty , \infty )$
  
C)  domain: $( \infty , 2 )$; range: $( \infty , \infty )$
  
D)  domain: $( \infty , 2 )$; range: $( \infty , \infty )$

Question 15

If the function is one-to-one, find its inverse. If not, write "not one-to-one."
-$$f ( x ) = \sqrt { x + 8 } , x \geq - 8$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \sqrt { x - 8 }$ 
B)  $f ^ { - 1 } ( x ) = x ^ { 2 } - 8 , x \geq 0$ 
C)  not a one-to-one 
D)  $f ^ { - 1 } ( x ) = ( x + 8 ) ^ { 2 }$ 
A)  $f ^ { - 1 } ( x ) = \sqrt { x - 8 }$ 
B)  $f ^ { - 1 } ( x ) = x ^ { 2 } - 8 , x \geq 0$ 
C)  not a one-to-one 
D)  $f ^ { - 1 } ( x ) = ( x + 8 ) ^ { 2 }$

Question 16

Choose the one alternative that best completes the statement or answers the question.
-The decay of $212 \mathrm { mg }$ of an isotope is given by $\mathrm { A } ( \mathrm { t } ) = 212 \mathrm { e } ^ { - 0.02 \mathrm { t } }$, where $\mathrm { t }$ is time in years. Find the amount left after 28 years.

Accepted Answer

A)  $119 \mathrm { mg }$ 
B)  $121 \mathrm { mg }$ 
C)  $208 \mathrm { mg }$ 
D)  $61 \mathrm { mg }$ 
A)  $119 \mathrm { mg }$ 
B)  $121 \mathrm { mg }$ 
C)  $208 \mathrm { mg }$ 
D)  $61 \mathrm { mg }$

Question 17

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth.
-$$e ^ { x } = 2 ^ { x + 1 }$$

Accepted Answer

A)  $\{ 6.52 \}$ 
B)  $\{ - 2.26 \}$ 
C)  $\{ 2.26 \}$ 
D)  $\{ 0.41 \}$ 
A)  $\{ 6.52 \}$ 
B)  $\{ - 2.26 \}$ 
C)  $\{ 2.26 \}$ 
D)  $\{ 0.41 \}$

Question 18

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent
positive real numbers.
-$$\log _ { 3 } \left( \frac { x ^ { 8 } y ^ { 4 } } { 6 } \right)$$

Accepted Answer

A)  $\left( 8 \log _ { 3 } x \right) \left( 4 \log _ { 3 } y \right) \div \log _ { 3 } 6$ 
B)  $8 \log _ { 3 } x + 4 \log _ { 3 } y - \log _ { 3 } 6$ 
C)  $8 \log _ { 3 } x + 4 \log _ { 3 } y + \log _ { 3 } 6$ 
D)  $\left( \log _ { 3 } x \right) ^ { 8 } + \left( \log _ { 3 } y \right) ^ { 4 } - \log _ { 3 } 6$ 
A)  $\left( 8 \log _ { 3 } x \right) \left( 4 \log _ { 3 } y \right) \div \log _ { 3 } 6$ 
B)  $8 \log _ { 3 } x + 4 \log _ { 3 } y - \log _ { 3 } 6$ 
C)  $8 \log _ { 3 } x + 4 \log _ { 3 } y + \log _ { 3 } 6$ 
D)  $\left( \log _ { 3 } x \right) ^ { 8 } + \left( \log _ { 3 } y \right) ^ { 4 } - \log _ { 3 } 6$

Question 19

Choose the one alternative that best completes the statement or answers the question.
-The half-life of an element is $5.4 \times 10 ^ { 9 } \mathrm { yr }$. How long does it take a sample of the element to decay to $\frac { 2 } { 3 }$ of its original mass? Use $A = A _ { 0 } ( 0.5 ) ^ { ( t / T ) }$ where $A _ { 0 }$ is the initial amount, $T$ is the half-life, and $t$ is the time. (Express results in scientific notation, rounded to the nearest hundredth.)

Accepted Answer

A)  $1.52 \times 10 ^ { 9 } \mathrm { yr }$ 
B)  $8.39 \times 10 ^ { 9 } \mathrm { yr }$ 
C)  $1.55 \times 10 ^ { 9 } \mathrm { yr }$ 
D)  $3.16 \times 10 ^ { 9 } \mathrm { yr }$ 
A)  $1.52 \times 10 ^ { 9 } \mathrm { yr }$ 
B)  $8.39 \times 10 ^ { 9 } \mathrm { yr }$ 
C)  $1.55 \times 10 ^ { 9 } \mathrm { yr }$ 
D)  $3.16 \times 10 ^ { 9 } \mathrm { yr }$

Question 20

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe
each inverse activity.
-setting a clock forward by one hour

Accepted Answer

A)  moving your schedule backward by one hour 
B)  leaving the clock as it is 
C)  setting the clock forward by two hours 
D)  setting the clock backward by one hour 
A)  moving your schedule backward by one hour 
B)  leaving the clock as it is 
C)  setting the clock forward by two hours 
D)  setting the clock backward by one hour

Determine the function value. -Suppose $f ( x ) = \log _ { a } x$ and $f ( 5 ) = 2$ . Find $f \left( \frac { 1 } { 25 } \right)$

Write an equivalent expression in exponential form. - $x = 3 ^ { \log _ { 3 } 14 }$

Solve the equation. Round to the nearest thousandth. - $53 x = 6 ( x + 1 )$

Solve the problem. -The population growth of an animal species is described by $\mathrm { F } ( \mathrm { t } ) = 400 \log ( 2 \mathrm { t } + 3 )$ where $\mathrm { t }$ is measured in months. Find the population of this species in an area 6 months after the species is introduced.

rite the word or phrase that best completes each statement or answers the question. -Write the algebraic equation which can be used to find the exact solution of $\log _ { 2 } 2 ( x + 4 ) =$ $\frac { 40 } { x - 10 }$

Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] , \text {as }$ needed. -Find $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ if the $\mathrm { pH } = 11$ .

Use properties of logarithms to evaluate the expression. -If $\mathrm { f } ( \mathrm { x } ) = \log 7 x$ , find $f \left( 7 ^ { 8 } \right)$

rite the word or phrase that best completes each statement or answers the question. -Without solving the inequality, what are the restrictions on $x$ if $\log _ { 3 } ( x - 4 ) - \log _ { 3 } ( x + 1 ) <$ 3? Explain.

Find the value. Give an approximation to four decimal places. - $\ln 295$

Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] , \text {as }$ needed. -Find the $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ if the $\mathrm { pH } = 2.3$ .

Graph the exponential function using transformations where appropriate. - $f(x)=-5^{x}$ $Graph the exponential function using transformations where appropriate. - f(x)=-5^{x}$

Graph the function. Give the domain and range. - $f ( x ) = \log _ { 1 / 2 } ( 2 - x )$ $Graph the function. Give the domain and range. - f ( x ) = \log _ { 1 / 2 } ( 2 - x )$

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $f ( x ) = \sqrt { x + 8 } , x \geq - 8$

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $e ^ { x } = 2 ^ { x + 1 }$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 3 } \left( \frac { x ^ { 8 } y ^ { 4 } } { 6 } \right)$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe each inverse activity. -setting a clock forward by one hour

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

Determine the function value. -Suppose f(x)=log⁡axf ( x ) = \log _ { a } xf(x)=loga​x and f(5)=2f ( 5 ) = 2f(5)=2 . Find f(125)f \left( \frac { 1 } { 25 } \right)f(251​)

Write an equivalent expression in exponential form. - x=3log⁡314x = 3 ^ { \log _ { 3 } 14 }x=3log3​14

Solve the equation. Round to the nearest thousandth. - 53x=6(x+1)53 x = 6 ( x + 1 )53x=6(x+1)

rite the word or phrase that best completes each statement or answers the question. -Write the algebraic equation which can be used to find the exact solution of log⁡22(x+4)=\log _ { 2 } 2 ( x + 4 ) =log2​2(x+4)= 40x−10\frac { 40 } { x - 10 }x−1040​

Use properties of logarithms to evaluate the expression. -If f(x)=log⁡7x\mathrm { f } ( \mathrm { x } ) = \log 7 xf(x)=log7x , find f(78)f \left( 7 ^ { 8 } \right)f(78)

rite the word or phrase that best completes each statement or answers the question. -Without solving the inequality, what are the restrictions on xxx if log⁡3(x−4)−log⁡3(x+1)<\log _ { 3 } ( x - 4 ) - \log _ { 3 } ( x + 1 ) <log3​(x−4)−log3​(x+1)< 3? Explain.

Find the value. Give an approximation to four decimal places. - ln⁡295\ln 295ln295

Graph the exponential function using transformations where appropriate. - f(x)=−5xf(x)=-5^{x}f(x)=−5x

Graph the function. Give the domain and range. - f(x)=log⁡1/2(2−x)f ( x ) = \log _ { 1 / 2 } ( 2 - x )f(x)=log1/2​(2−x)

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - f(x)=x+8,x≥−8f ( x ) = \sqrt { x + 8 } , x \geq - 8f(x)=x+8​,x≥−8

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - ex=2x+1e ^ { x } = 2 ^ { x + 1 }ex=2x+1

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - log⁡3(x8y46)\log _ { 3 } \left( \frac { x ^ { 8 } y ^ { 4 } } { 6 } \right)log3​(6x8y4​)