Question 1

Give the domain and range.
-$f(x)=\log _{6} x^{6}$

Accepted Answer

A)  domain: $( \infty , 0 ) \cup ( 0 , \infty )$; range: $( \infty , \infty )$
  
B)  domain: $( 0 , \infty )$; range: $( \infty , \infty )$
  
C)  domain: $( 0 , \infty )$; range: $( \infty , \infty )$
  
D)  domain: $( * , 0 ) \cup ( 0 , \infty )$; range: $( - \infty , \infty )$
  
A)  domain: $( \infty , 0 ) \cup ( 0 , \infty )$; range: $( \infty , \infty )$
  
B)  domain: $( 0 , \infty )$; range: $( \infty , \infty )$
  
C)  domain: $( 0 , \infty )$; range: $( \infty , \infty )$
  
D)  domain: $( * , 0 ) \cup ( 0 , \infty )$; range: $( - \infty , \infty )$

Question 2

Use the change of base rule to find the logarithm to four decimal places.
-$\log _ { 7 } 3$

Accepted Answer

A)  0.5646 
B)  1.1292 
C)  1.7712 
D)  -0.5646 
A)  0.5646 
B)  1.1292 
C)  1.7712 
D)  -0.5646

Question 3

Solve the following graphically. If necessary, round answers to the nearest thousandth.
-$$\mathrm { e } ^ { \mathrm { x } } + \ln 3 = 9 \mathrm { e } ^ { \mathrm { x } }$$

Accepted Answer

A)  $\{ - 2.209 \}$ 
B)  $\{ - 1.985 \}$ 
C)  $\{ 1.099 \}$ 
D)  $\{ - 3.970 \}$ 
A)  $\{ - 2.209 \}$ 
B)  $\{ - 1.985 \}$ 
C)  $\{ 1.099 \}$ 
D)  $\{ - 3.970 \}$

Question 4

Write an equivalent expression in exponential form.
-$\log _ { x } 16 = - 2$

Accepted Answer

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

Question 5

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ .
-$$\left( \log _ { a } x - \log _ { a } y \right) + 5 \log _ { a } z$$

Accepted Answer

A)  $\log _ { a } \left( \frac { x z ^ { 5 } } { y } \right)$ 
B)  $\log _ { a } \left( \frac { 5 x z } { y } \right)$ 
C)  $\log _ { a } \left( x + z ^ { 5 } - y \right)$ 
D)  $\log _ { a } \left( \frac { x } { z ^ { 5 } y } \right)$ 
A)  $\log _ { a } \left( \frac { x z ^ { 5 } } { y } \right)$ 
B)  $\log _ { a } \left( \frac { 5 x z } { y } \right)$ 
C)  $\log _ { a } \left( x + z ^ { 5 } - y \right)$ 
D)  $\log _ { a } \left( \frac { x } { z ^ { 5 } y } \right)$

Question 6

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers.
-$\log _ { 19 } \left( \frac { 14 \mathrm {~m} } { \mathrm {~s} } \right)$

Accepted Answer

A)  $\log _ { 19 } 14 + \log 19 m - \log 19 s$ 
B)  $\left( \log _ { 19 } 14 \right) ( \log 19 \mathrm {~m} ) - \log _ { 19 } \mathrm {~s}$ 
C)  $\frac { \log 1914 + \log 19 \mathrm {~m} } { \log 19 \mathrm {~s} }$ 
D)  $\log 1914 - \log 19 \mathrm {~m} - \log 19 \mathrm {~s}$ 
A)  $\log _ { 19 } 14 + \log 19 m - \log 19 s$ 
B)  $\left( \log _ { 19 } 14 \right) ( \log 19 \mathrm {~m} ) - \log _ { 19 } \mathrm {~s}$ 
C)  $\frac { \log 1914 + \log 19 \mathrm {~m} } { \log 19 \mathrm {~s} }$ 
D)  $\log 1914 - \log 19 \mathrm {~m} - \log 19 \mathrm {~s}$

Question 7

Evaluate the logarithm.
-$\log _ { 10 } 0.0001$

Accepted Answer

A)  $- 0.0004$ 
B)  $0.0001$ 
C)  $- 4$ 
D)  4 
A)  $- 0.0004$ 
B)  $0.0001$ 
C)  $- 4$ 
D)  4

Question 8

Find the future value.
-$19,691 invested for 9 years at 3% compounded semiannually

Accepted Answer

A)  $25,785.82 
B)  $25,768.54 
C)  $25,742.84 
D)  $25,692.29 
A)  $25,785.82 
B)  $25,768.54 
C)  $25,742.84 
D)  $25,692.29

Question 9

In the formula N $\mathrm { N } = \mathrm { I } \mathrm { e } ^ { \mathrm { kt } }$ , N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. How long will it take for the population of a
Certain country to double if its annual growth rate is 5.7%? Round to the nearest year.

Accepted Answer

A)  35 yr 
B)  12 yr 
C)  5 yr 
D)  1 yr 
A)  35 yr 
B)  12 yr 
C)  5 yr 
D)  1 yr

Question 10

Use properties of logarithms to evaluate the expression.
-$\log _ { 10 } ( 0.01 ) ^ { 7 }$

Accepted Answer

A)  $\frac { 1 } { 14 }$ 
B)  $- 49$ 
C)  $\frac { 1 } { 49 }$ 
D)  $- 14$ 
A)  $\frac { 1 } { 14 }$ 
B)  $- 49$ 
C)  $\frac { 1 } { 49 }$ 
D)  $- 14$

Question 11

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

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \frac { 8 + 8 x } { x }$ 
B)  $f ^ { - 1 } ( x ) = \frac { x } { 8 + 8 x }$ 
C)  not a one-to-one 
D)  $f ^ { - 1 } ( x ) = \frac { - 8 x + 8 } { x }$ 
A)  $f ^ { - 1 } ( x ) = \frac { 8 + 8 x } { x }$ 
B)  $f ^ { - 1 } ( x ) = \frac { x } { 8 + 8 x }$ 
C)  not a one-to-one 
D)  $f ^ { - 1 } ( x ) = \frac { - 8 x + 8 } { x }$

Question 12

Write in logarithmic form.
-$$10 ^ { - 2 } = 0.01$$

Accepted Answer

A)  $\log _ { 10 } 0.01 = - 2$ 
B)  $\log _ { 2 } - 2 = 0.10$ 
C)  $\log _ { 10 } - 2 = 0.01$ 
D)  $\log _ { 2 } 0.10 = - 2$ 
A)  $\log _ { 10 } 0.01 = - 2$ 
B)  $\log _ { 2 } - 2 = 0.10$ 
C)  $\log _ { 10 } - 2 = 0.01$ 
D)  $\log _ { 2 } 0.10 = - 2$

Question 13

Solve for the indicated variable.
-$\mathrm { I } - \mathrm { I } _ { 0 } = \left( \mathrm { I } _ { 1 } - \mathrm { I } _ { 0 } \right) 10 ^ { - \mathrm { kt } }$, for $\mathrm { t }$

Accepted Answer

A)  $\mathrm { t } = - \frac { 1 } { \mathrm { k } } \log \left( \mathrm { I } - \mathrm { I } _ { 1 } \right)$ 
B)  $t = - \frac { 1 } { k } \log \frac { \mathrm { I } } { \mathrm { I } }$ 
C)  $t = - \frac { I - I _ { 0 } } { k \left( I _ { 1 } - I _ { 0 } \right) }$ 
D)  $t = - \frac { 1 } { k } \log \frac { \mathrm { I } - \mathrm { I } _ { 0 } } { \mathrm { I } _ { 1 } - \mathrm { I } _ { 0 } }$ 
A)  $\mathrm { t } = - \frac { 1 } { \mathrm { k } } \log \left( \mathrm { I } - \mathrm { I } _ { 1 } \right)$ 
B)  $t = - \frac { 1 } { k } \log \frac { \mathrm { I } } { \mathrm { I } }$ 
C)  $t = - \frac { I - I _ { 0 } } { k \left( I _ { 1 } - I _ { 0 } \right) }$ 
D)  $t = - \frac { 1 } { k } \log \frac { \mathrm { I } - \mathrm { I } _ { 0 } } { \mathrm { I } _ { 1 } - \mathrm { I } _ { 0 } }$

Question 14

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ .
-$4 \log _ { a } m - \log _ { a } n$

Accepted Answer

A)  $\log _ { a } \left( \frac { m ^ { 4 } } { n } \right)$ 
B)  $\log _ { a } \left( m ^ { 4 } - n \right)$ 
C)  $\log _ { a } \left( \frac { 4 m } { n } \right)$ 
D)  $\log _ { a } m ^ { 4 } \div \log _ { a } n$ 
A)  $\log _ { a } \left( \frac { m ^ { 4 } } { n } \right)$ 
B)  $\log _ { a } \left( m ^ { 4 } - n \right)$ 
C)  $\log _ { a } \left( \frac { 4 m } { n } \right)$ 
D)  $\log _ { a } m ^ { 4 } \div \log _ { a } n$

Question 15

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers.
-$\log b \sqrt [ 3 ] { \frac { x ^ { 6 } } { y ^ { 2 } z ^ { 8 } } }$

Accepted Answer

A)  $2 \log _ { b } x \div \left( \frac { 2 } { 3 } \log _ { b y } \cdot \frac { 8 } { 3 } \log _ { b } z \right)$ 
B)  $\frac { 1 } { 3 } \log _ { b } x ^ { 6 } - \log _ { b } y ^ { 2 } - \log _ { b } z ^ { 8 }$ 
C)  $6 \log _ { b } x - 2 \log _ { b } y - 8 \log _ { b } z$ 
D)  $2 \log _ { b } x - \frac { 2 } { 3 } \log _ { b } y - \frac { 8 } { 3 } \log _ { b } z$ 
A)  $2 \log _ { b } x \div \left( \frac { 2 } { 3 } \log _ { b y } \cdot \frac { 8 } { 3 } \log _ { b } z \right)$ 
B)  $\frac { 1 } { 3 } \log _ { b } x ^ { 6 } - \log _ { b } y ^ { 2 } - \log _ { b } z ^ { 8 }$ 
C)  $6 \log _ { b } x - 2 \log _ { b } y - 8 \log _ { b } z$ 
D)  $2 \log _ { b } x - \frac { 2 } { 3 } \log _ { b } y - \frac { 8 } { 3 } \log _ { b } z$

Question 16

Determine whether the statement is true or false.
-Any function of the form f(x) $f ( x ) = x ^ { n }$ where n is an odd integer, has an inverse.

Accepted Answer

A) True 
 B)False

Question 17

An artifact is discovered at a certain site. If it has 53% of the carbon-14 it originally contained, what is the approximate age of the artifact to the nearest year? (carbon-14 decays at the rate of 0.0125% annually.)

Accepted Answer

A)  2206 yr 
B)  4240 yr 
C)  5079 yr 
D)  3760 yr 
A)  2206 yr 
B)  4240 yr 
C)  5079 yr 
D)  3760 yr

Question 18

If the function is one-to-one, find its inverse. If not, write "not one-to-one."
-$$\{ ( 12,1 ) , ( - 10,1 ) , ( 11 , - 14 ) \}$$

Accepted Answer

A)  $\{ ( 1,12 ) , ( 1 , - 10 ) , ( - 14,11 ) \}$ 
B)  $\{ ( 12,1 ) , ( 1 , - 10 ) , ( - 14,11 ) \}$ 
C)  $\{ ( 1,12 ) , ( 11 , - 10 ) , ( - 14,1 ) \}$ 
D)  not one-to-one 
A)  $\{ ( 1,12 ) , ( 1 , - 10 ) , ( - 14,11 ) \}$ 
B)  $\{ ( 12,1 ) , ( 1 , - 10 ) , ( - 14,11 ) \}$ 
C)  $\{ ( 1,12 ) , ( 11 , - 10 ) , ( - 14,1 ) \}$ 
D)  not one-to-one

Question 19

Given $\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$ find the logarithm without using a calculator.
-$\log _ { 10 } 6$

Accepted Answer

A)  0.9542 
B)  0.903 
C)  0.7781 
D)  0.1436 
A)  0.9542 
B)  0.903 
C)  0.7781 
D)  0.1436

Question 20

Solve the following graphically. If necessary, round answers to the nearest thousandth.
-$$\mathrm { e } ^ { \mathrm { x } + \ln 8 } = 10 \mathrm { e } ^ { \mathrm { x } }$$

Accepted Answer

A)  $\{ 2.079 \}$ 
B)  $\{ 7.921 \}$ 
C)  $\varnothing$ 
D)  $\{ 0.208 \}$ 
A)  $\{ 2.079 \}$ 
B)  $\{ 7.921 \}$ 
C)  $\varnothing$ 
D)  $\{ 0.208 \}$

Give the domain and range. - $f(x)=\log _{6} x^{6}$ $Give the domain and range. - f(x)=\log _{6} x^{6}$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 7 } 3$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\mathrm { e } ^ { \mathrm { x } } + \ln 3 = 9 \mathrm { e } ^ { \mathrm { x } }$

Write an equivalent expression in exponential form. - $\log _ { x } 16 = - 2$

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ . - $\left( \log _ { a } x - \log _ { a } y \right) + 5 \log _ { a } z$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 19 } \left( \frac { 14 \mathrm {~m} } { \mathrm {~s} } \right)$

Evaluate the logarithm. - $\log _ { 10 } 0.0001$

Find the future value. -$19,691 invested for 9 years at 3% compounded semiannually

Use properties of logarithms to evaluate the expression. - $\log _ { 10 } ( 0.01 ) ^ { 7 }$

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

Write in logarithmic form. - $10 ^ { - 2 } = 0.01$

Solve for the indicated variable. - $\mathrm { I } - \mathrm { I } _ { 0 } = \left( \mathrm { I } _ { 1 } - \mathrm { I } _ { 0 } \right) 10 ^ { - \mathrm { kt } }$ , for $\mathrm { t }$

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ . - $4 \log _ { a } m - \log _ { a } n$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log b \sqrt [ 3 ] { \frac { x ^ { 6 } } { y ^ { 2 } z ^ { 8 } } }$

Determine whether the statement is true or false. -Any function of the form f(x) $f ( x ) = x ^ { n }$ where n is an odd integer, has an inverse.

An artifact is discovered at a certain site. If it has 53% of the carbon-14 it originally contained, what is the approximate age of the artifact to the nearest year? (carbon-14 decays at the rate of 0.0125% annually.)

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $\{ ( 12,1 ) , ( - 10,1 ) , ( 11 , - 14 ) \}$

Given $\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$ find the logarithm without using a calculator. - $\log _ { 10 } 6$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\mathrm { e } ^ { \mathrm { x } + \ln 8 } = 10 \mathrm { e } ^ { \mathrm { x } }$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Systems and Matrices

Arithmetic Sequence: Common Difference and First n Terms

Filters

Exam 5: Inverse, Exponential, and Logarithmic Functions

Give the domain and range. - f(x)=log⁡6x6f(x)=\log _{6} x^{6}f(x)=log6​x6

Use the change of base rule to find the logarithm to four decimal places. - log⁡73\log _ { 7 } 3log7​3

Solve the following graphically. If necessary, round answers to the nearest thousandth. - ex+ln⁡3=9ex\mathrm { e } ^ { \mathrm { x } } + \ln 3 = 9 \mathrm { e } ^ { \mathrm { x } }ex+ln3=9ex

Write an equivalent expression in exponential form. - log⁡x16=−2\log _ { x } 16 = - 2logx​16=−2

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - log⁡19(14 m s)\log _ { 19 } \left( \frac { 14 \mathrm {~m} } { \mathrm {~s} } \right)log19​( s14 m​)

Evaluate the logarithm. - log⁡100.0001\log _ { 10 } 0.0001log10​0.0001

Find the future value. -$19,691 invested for 9 years at 3% compounded semiannually

Use properties of logarithms to evaluate the expression. - log⁡10(0.01)7\log _ { 10 } ( 0.01 ) ^ { 7 }log10​(0.01)7

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - f(x)=8x+8f ( x ) = \frac { 8 } { x + 8 }f(x)=x+88​

Write in logarithmic form. - 10−2=0.0110 ^ { - 2 } = 0.0110−2=0.01

Solve for the indicated variable. - I−I0=(I1−I0)10−kt\mathrm { I } - \mathrm { I } _ { 0 } = \left( \mathrm { I } _ { 1 } - \mathrm { I } _ { 0 } \right) 10 ^ { - \mathrm { kt } }I−I0​=(I1​−I0​)10−kt , for t\mathrm { t }t

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 . - 4log⁡am−log⁡an4 \log _ { a } m - \log _ { a } n4loga​m−loga​n

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - log⁡bx6y2z83\log b \sqrt [ 3 ] { \frac { x ^ { 6 } } { y ^ { 2 } z ^ { 8 } } }logb3y2z8x6​​

Determine whether the statement is true or false. -Any function of the form f(x) f(x)=xnf ( x ) = x ^ { n }f(x)=xn where n is an odd integer, has an inverse.

An artifact is discovered at a certain site. If it has 53% of the carbon-14 it originally contained, what is the approximate age of the artifact to the nearest year? (carbon-14 decays at the rate of 0.0125% annually.)

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - {(12,1),(−10,1),(11,−14)}\{ ( 12,1 ) , ( - 10,1 ) , ( 11 , - 14 ) \}{(12,1),(−10,1),(11,−14)}

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⁡106\log _ { 10 } 6log10​6

Solve the following graphically. If necessary, round answers to the nearest thousandth. - ex+ln⁡8=10ex\mathrm { e } ^ { \mathrm { x } + \ln 8 } = 10 \mathrm { e } ^ { \mathrm { x } }ex+ln8=10ex

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Systems and Matrices

Arithmetic Sequence: Common Difference and First n Terms

Filters

Give the domain and range. - $f(x)=\log _{6} x^{6}$ $Give the domain and range. - f(x)=\log _{6} x^{6}$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 7 } 3$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\mathrm { e } ^ { \mathrm { x } } + \ln 3 = 9 \mathrm { e } ^ { \mathrm { x } }$

Write an equivalent expression in exponential form. - $\log _ { x } 16 = - 2$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 19 } \left( \frac { 14 \mathrm {~m} } { \mathrm {~s} } \right)$

Evaluate the logarithm. - $\log _ { 10 } 0.0001$

Use properties of logarithms to evaluate the expression. - $\log _ { 10 } ( 0.01 ) ^ { 7 }$

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

Write in logarithmic form. - $10 ^ { - 2 } = 0.01$

Solve for the indicated variable. - $\mathrm { I } - \mathrm { I } _ { 0 } = \left( \mathrm { I } _ { 1 } - \mathrm { I } _ { 0 } \right) 10 ^ { - \mathrm { kt } }$ , for $\mathrm { t }$

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ . - $4 \log _ { a } m - \log _ { a } n$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log b \sqrt [ 3 ] { \frac { x ^ { 6 } } { y ^ { 2 } z ^ { 8 } } }$

Determine whether the statement is true or false. -Any function of the form f(x) $f ( x ) = x ^ { n }$ where n is an odd integer, has an inverse.

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $\{ ( 12,1 ) , ( - 10,1 ) , ( 11 , - 14 ) \}$

Given $\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$ find the logarithm without using a calculator. - $\log _ { 10 } 6$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\mathrm { e } ^ { \mathrm { x } + \ln 8 } = 10 \mathrm { e } ^ { \mathrm { x } }$