Question 1

Solve the problem.
-If $f ( x ) = 25 ^ { x }$, find $f [ \log 25 ( \ln 25 ) ]$.

Accepted Answer

A)  0 
B)  $25 \ln 25$ 
C)  $\ln 25$ 
D)  $\log _ { 25 } 25$ 
A)  0 
B)  $25 \ln 25$ 
C)  $\ln 25$ 
D)  $\log _ { 25 } 25$

Question 2

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers.
-$\log _ { 3 } \left( \frac { x ^ { 4 } y ^ { 5 } } { 6 } \right)$

Accepted Answer

A)  $4 \log _ { 3 } x + 5 \log _ { 3 } y - \log _ { 3 } 6$ 
B)  $\left( \log _ { 3 } x \right) ^ { 4 } + \left( \log _ { 3 } y \right) ^ { 5 } - \log _ { 3 } 6$ 
C)  $\left( 4 \log _ { 3 } x \right) \left( 5 \log _ { 3 } y \right) \div \log _ { 3 } 6$ 
D)  $4 \log _ { 3 } x + 5 \log 3 y + \log _ { 3 } 6$ 
A)  $4 \log _ { 3 } x + 5 \log _ { 3 } y - \log _ { 3 } 6$ 
B)  $\left( \log _ { 3 } x \right) ^ { 4 } + \left( \log _ { 3 } y \right) ^ { 5 } - \log _ { 3 } 6$ 
C)  $\left( 4 \log _ { 3 } x \right) \left( 5 \log _ { 3 } y \right) \div \log _ { 3 } 6$ 
D)  $4 \log _ { 3 } x + 5 \log 3 y + \log _ { 3 } 6$

Question 3

Find the value. Give an approximation to four decimal places.
-ln 0.000772

Accepted Answer

A)  -3.1124 
B)  3.1124 
C)  -7.1665 
D)  7.1665 
A)  -3.1124 
B)  3.1124 
C)  -7.1665 
D)  7.1665

Question 4

Write an equivalent expression in exponential form.
-$\log _ { 3 } \left( \frac { 1 } { 9 } \right) = - 2$

Accepted Answer

A)  $2 ^ { 3 } = \frac { 1 } { 9 }$ 
B)  $3 ^ { 1 / 9 } = 2$ 
C)  $3 ^ { - 2 } = \frac { 1 } { 9 }$ 
D)  $\left( \frac { 1 } { 9 } \right) ^ { 2 } = 3$ 
A)  $2 ^ { 3 } = \frac { 1 } { 9 }$ 
B)  $3 ^ { 1 / 9 } = 2$ 
C)  $3 ^ { - 2 } = \frac { 1 } { 9 }$ 
D)  $\left( \frac { 1 } { 9 } \right) ^ { 2 } = 3$

Question 5

Given $$\log _ { 10 } 2 = 0.3010 \text { and } \log _ { 10 } 3 = 0.4771 \text {, }$$ find the logarithm without using a calculator.
-$$\log _ { 10 } \frac { 9 } { 8 }$$

Accepted Answer

A)  0.1992 
B)  0.0512 
C)  2.0333 
D)  0.8293 
A)  0.1992 
B)  0.0512 
C)  2.0333 
D)  0.8293

Question 6

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

Accepted Answer

A)  $\{ 80 \}$ 
B)  $\left\{ \frac { 67 } { 4 } \right\}$ 
C)  $\{ 4 \sqrt { 5 } \}$ 
D)  $\left\{ \frac { 64 } { 3 } \right\}$ 
A)  $\{ 80 \}$ 
B)  $\left\{ \frac { 67 } { 4 } \right\}$ 
C)  $\{ 4 \sqrt { 5 } \}$ 
D)  $\left\{ \frac { 64 } { 3 } \right\}$

Question 7

Graph the exponential function using transformations where appropriate.
-$f(x)=4^{x-5}$

Accepted Answer

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

Question 8

Provide an appropriate response.
-Use the properties of exponents to write the function of the form $f ( t ) = k a t$, where $k$ is a constant. $f ( t ) = 3 ^ { 3 t } + 2$

Accepted Answer

A)  $f ( t ) = 9 \cdot 3^{3 t}$ 
B)  $f ( t ) = 9 \cdot 27^ t$ 
C)  $f ( t ) = 27^ t + 9$ 
D)  $f ( t ) = 27 \cdot 9^ t$ 
A)  $f ( t ) = 9 \cdot 3^{3 t}$ 
B)  $f ( t ) = 9 \cdot 27^ t$ 
C)  $f ( t ) = 27^ t + 9$ 
D)  $f ( t ) = 27 \cdot 9^ t$

Question 9

Solve the problem. Round your answer to the nearest tenth, when appropriate. Use the formula $$\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$$ as needed.
-Find the $\mathrm { pH }$ if $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right] = 5.9 \times 10 ^ { - 4 }$.

Accepted Answer

A)  $3.6$ 
B)  $4.8$ 
C)  $3.2$ 
D)  $- 4.8$ 
A)  $3.6$ 
B)  $4.8$ 
C)  $3.2$ 
D)  $- 4.8$

Question 10

Write the word or phrase that best completes each statement or answers the question.
-$$\text { With the function } \mathrm { f } ( \mathrm { x } ) = \log _ { \mathrm { a } } x \text {, why can't } x \text { be less than } 0 \text { ? }$$

Accepted Answer

Answers may vary. One possibility: The f

Question 11

Write the word or phrase that best completes each statement or answers the question.
-How long will it take a sample of radioactive substance to decay to half of its original amount, if it decays according to the function $\mathrm { A } ( \mathrm { t } ) = 450 \mathrm { e } ^ { - .168 \mathrm { t } }$, where $\mathrm { t }$ is the time in years? Round your answer to the nearest hundredth year.

Accepted Answer

A)  $4.13 \mathrm { yr }$ 
B)  $36.36 \mathrm { yr }$ 
C)  $75.60 \mathrm { yr }$ 
D)  $40.49 \mathrm { yr }$ 
A)  $4.13 \mathrm { yr }$ 
B)  $36.36 \mathrm { yr }$ 
C)  $75.60 \mathrm { yr }$ 
D)  $40.49 \mathrm { yr }$

Question 12

Provide an appropriate response.
-$$\text { Explain how the graph of } y = 3 ^ { x } - 3 + 1 \text { can be obtained from the graph of } y = 3 ^ { x }$$

Accepted Answer

The graph is shifted

Question 13

Solve the equation.
-$e ^ { x - 1 } = \left( \frac { 1 } { e ^ { 2 } } \right) ^ { x + 5 }$

Accepted Answer

A)  $\{ - 11 \}$ 
B)  $\{ 2 \}$ 
C)  $\{ - 3 \}$ 
D)  $\{ - 6 \}$ 
A)  $\{ - 11 \}$ 
B)  $\{ 2 \}$ 
C)  $\{ - 3 \}$ 
D)  $\{ - 6 \}$

Question 14

If the function is one-to-one, find its inverse. If not, write ʺnot one-to-one.ʺ
-$$f ( x ) = 7 x ^ { 3 } + 6$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x + 6 } { 7 } }$ 
B)  not a one-to-one 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt [ 3 ] { \frac { \mathrm { x } - 6 } { 7 } }$ 
D)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x } { 7 } } - 6$ 
A)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x + 6 } { 7 } }$ 
B)  not a one-to-one 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sqrt [ 3 ] { \frac { \mathrm { x } - 6 } { 7 } }$ 
D)  $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x } { 7 } } - 6$

Question 15

Solve the following graphically. If necessary, round answers to the nearest thousandth.
-$$\mathrm { e } ^ { \mathrm { x } } + \ln 3 = 4 \mathrm { e } ^ { \mathrm { x } }$$

Accepted Answer

A)  $\{ - 1.005 \}$ 
B)  $\{ 1.099 \}$ 
C)  $\{ - 2.010 \}$ 
D)  $\{ - 1.515 \}$ 
A)  $\{ - 1.005 \}$ 
B)  $\{ 1.099 \}$ 
C)  $\{ - 2.010 \}$ 
D)  $\{ - 1.515 \}$

Question 16

For the function as defined that is one-to-one, graph f and f-1 on the same axes.
-$f(x)=x^{3}+4$
 
A)
 
B)
 
C)
 
D)

Accepted Answer

The answer of For the function as defined that is...

Question 17

Graph the function.
-$f(x)=\left(\frac{1}{2}\right)^{x}$

Accepted Answer

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

Question 18

Provide an appropriate response.
-$$\text { Why can't } \mathrm { y } = 2 ^ { \mathrm { x } } \text { have an } \mathrm { x } \text {-intercept? }$$

Accepted Answer

Because no

Question 19

For the function as defined that is one-to-one, graph f and f-1 on the same axes.
-$$f ( x ) = \frac { 4 } { 3 } x + 4$$

Accepted Answer

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

Question 20

Find the domain and range of the inverse of the given function.
-$$f ( x ) = 7 - x ^ { 2 } ; x \geq 0$$

Accepted Answer

A)  Domain: all real numbers; range: $[ - 7 , \infty )$ 
B)  Domain: all real numbers; range: $( - \infty , 7 ]$ 
C)  Domain: $( - \infty , 7 ]$; range: range: $[ 0 , \infty )$ 
D)  Domain and range: all real numbers 
A)  Domain: all real numbers; range: $[ - 7 , \infty )$ 
B)  Domain: all real numbers; range: $( - \infty , 7 ]$ 
C)  Domain: $( - \infty , 7 ]$; range: range: $[ 0 , \infty )$ 
D)  Domain and range: all real numbers

Solve the problem. -If $f ( x ) = 25 ^ { x }$ , find $f [ \log 25 ( \ln 25 ) ]$ .

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 3 } \left( \frac { x ^ { 4 } y ^ { 5 } } { 6 } \right)$

Find the value. Give an approximation to four decimal places. -ln 0.000772

Write an equivalent expression in exponential form. - $\log _ { 3 } \left( \frac { 1 } { 9 } \right) = - 2$

Given $\log _ { 10 } 2 = 0.3010 \text { and } \log _ { 10 } 3 = 0.4771 \text {, }$ find the logarithm without using a calculator. - $\log _ { 10 } \frac { 9 } { 8 }$

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

Graph the exponential function using transformations where appropriate. - $f(x)=4^{x-5}$ $Graph the exponential function using transformations where appropriate. - f(x)=4^{x-5}$

Provide an appropriate response. -Use the properties of exponents to write the function of the form $f ( t ) = k a t$ , where $k$ is a constant. $f ( t ) = 3 ^ { 3 t } + 2$

Write the word or phrase that best completes each statement or answers the question. - $\text { With the function } \mathrm { f } ( \mathrm { x } ) = \log _ { \mathrm { a } } x \text {, why can't } x \text { be less than } 0 \text { ? }$

Provide an appropriate response. - $\text { Explain how the graph of } y = 3 ^ { x } - 3 + 1 \text { can be obtained from the graph of } y = 3 ^ { x }$

Solve the equation. - $e ^ { x - 1 } = \left( \frac { 1 } { e ^ { 2 } } \right) ^ { x + 5 }$

If the function is one-to-one, find its inverse. If not, write ʺnot one-to-one.ʺ - $f ( x ) = 7 x ^ { 3 } + 6$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\mathrm { e } ^ { \mathrm { x } } + \ln 3 = 4 \mathrm { e } ^ { \mathrm { x } }$

Graph the function. - $f(x)=\left(\frac{1}{2}\right)^{x}$ $Graph the function. - f(x)=\left(\frac{1}{2}\right)^{x}$

Provide an appropriate response. - $\text { Why can't } \mathrm { y } = 2 ^ { \mathrm { x } } \text { have an } \mathrm { x } \text {-intercept? }$

For the function as defined that is one-to-one, graph f and f-1 on the same axes. - $f ( x ) = \frac { 4 } { 3 } x + 4$ $For the function as defined that is one-to-one, graph f and f-1 on the same axes. - f ( x ) = \frac { 4 } { 3 } x + 4$

Find the domain and range of the inverse of the given function. - $f ( x ) = 7 - x ^ { 2 } ; x \geq 0$

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

Review of Basic Concepts

Filters

Exam 4: Inverse, Exponential, and Logarithmic Functions

Solve the problem. -If f(x)=25xf ( x ) = 25 ^ { x }f(x)=25x , find f[log⁡25(ln⁡25)]f [ \log 25 ( \ln 25 ) ]f[log25(ln25)] .

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - log⁡3(x4y56)\log _ { 3 } \left( \frac { x ^ { 4 } y ^ { 5 } } { 6 } \right)log3​(6x4y5​)

Find the value. Give an approximation to four decimal places. -ln 0.000772

Write an equivalent expression in exponential form. - log⁡3(19)=−2\log _ { 3 } \left( \frac { 1 } { 9 } \right) = - 2log3​(91​)=−2

Given log⁡102=0.3010 and log⁡103=0.4771, \log _ { 10 } 2 = 0.3010 \text { and } \log _ { 10 } 3 = 0.4771 \text {, }log10​2=0.3010 and log10​3=0.4771, find the logarithm without using a calculator. - log⁡1098\log _ { 10 } \frac { 9 } { 8 }log10​89​

Solve the equation and express the solution in exact form. - log⁡4(x+4)+log⁡4(x−4)=3\log _ { 4 } ( x + 4 ) + \log _ { 4 } ( x - 4 ) = 3log4​(x+4)+log4​(x−4)=3

Graph the exponential function using transformations where appropriate. - f(x)=4x−5f(x)=4^{x-5}f(x)=4x−5

Provide an appropriate response. -Use the properties of exponents to write the function of the form f(t)=katf ( t ) = k a tf(t)=kat , where kkk is a constant. f(t)=33t+2f ( t ) = 3 ^ { 3 t } + 2f(t)=33t+2

Provide an appropriate response. - Explain how the graph of y=3x−3+1 can be obtained from the graph of y=3x\text { Explain how the graph of } y = 3 ^ { x } - 3 + 1 \text { can be obtained from the graph of } y = 3 ^ { x } Explain how the graph of y=3x−3+1 can be obtained from the graph of y=3x

Solve the equation. - ex−1=(1e2)x+5e ^ { x - 1 } = \left( \frac { 1 } { e ^ { 2 } } \right) ^ { x + 5 }ex−1=(e21​)x+5

If the function is one-to-one, find its inverse. If not, write ʺnot one-to-one.ʺ - f(x)=7x3+6f ( x ) = 7 x ^ { 3 } + 6f(x)=7x3+6

Solve the following graphically. If necessary, round answers to the nearest thousandth. - ex+ln⁡3=4ex\mathrm { e } ^ { \mathrm { x } } + \ln 3 = 4 \mathrm { e } ^ { \mathrm { x } }ex+ln3=4ex

For the function as defined that is one-to-one, graph f and f-1 on the same axes. - f(x)=x3+4f(x)=x^{3}+4f(x)=x3+4 A) B) C) D)

Graph the function. - f(x)=(12)xf(x)=\left(\frac{1}{2}\right)^{x}f(x)=(21​)x

Provide an appropriate response. - Why can’t y=2x have an x-intercept? \text { Why can't } \mathrm { y } = 2 ^ { \mathrm { x } } \text { have an } \mathrm { x } \text {-intercept? } Why can’t y=2x have an x-intercept?

For the function as defined that is one-to-one, graph f and f-1 on the same axes. - f(x)=43x+4f ( x ) = \frac { 4 } { 3 } x + 4f(x)=34​x+4

Find the domain and range of the inverse of the given function. - f(x)=7−x2;x≥0f ( x ) = 7 - x ^ { 2 } ; x \geq 0f(x)=7−x2;x≥0

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

Review of Basic Concepts

Filters

Solve the problem. -If $f ( x ) = 25 ^ { x }$ , find $f [ \log 25 ( \ln 25 ) ]$ .

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 3 } \left( \frac { x ^ { 4 } y ^ { 5 } } { 6 } \right)$

Write an equivalent expression in exponential form. - $\log _ { 3 } \left( \frac { 1 } { 9 } \right) = - 2$

Given $\log _ { 10 } 2 = 0.3010 \text { and } \log _ { 10 } 3 = 0.4771 \text {, }$ find the logarithm without using a calculator. - $\log _ { 10 } \frac { 9 } { 8 }$

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

Graph the exponential function using transformations where appropriate. - $f(x)=4^{x-5}$ $Graph the exponential function using transformations where appropriate. - f(x)=4^{x-5}$

Provide an appropriate response. -Use the properties of exponents to write the function of the form $f ( t ) = k a t$ , where $k$ is a constant. $f ( t ) = 3 ^ { 3 t } + 2$

Provide an appropriate response. - $\text { Explain how the graph of } y = 3 ^ { x } - 3 + 1 \text { can be obtained from the graph of } y = 3 ^ { x }$

Solve the equation. - $e ^ { x - 1 } = \left( \frac { 1 } { e ^ { 2 } } \right) ^ { x + 5 }$

If the function is one-to-one, find its inverse. If not, write ʺnot one-to-one.ʺ - $f ( x ) = 7 x ^ { 3 } + 6$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\mathrm { e } ^ { \mathrm { x } } + \ln 3 = 4 \mathrm { e } ^ { \mathrm { x } }$

Graph the function. - $f(x)=\left(\frac{1}{2}\right)^{x}$ $Graph the function. - f(x)=\left(\frac{1}{2}\right)^{x}$

Provide an appropriate response. - $\text { Why can't } \mathrm { y } = 2 ^ { \mathrm { x } } \text { have an } \mathrm { x } \text {-intercept? }$

For the function as defined that is one-to-one, graph f and f-1 on the same axes. - $f ( x ) = \frac { 4 } { 3 } x + 4$ $For the function as defined that is one-to-one, graph f and f-1 on the same axes. - f ( x ) = \frac { 4 } { 3 } x + 4$

Find the domain and range of the inverse of the given function. - $f ( x ) = 7 - x ^ { 2 } ; x \geq 0$