Question 1

Given $$\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$$ find the logarithm without using a calculator.
-$\log _ { 10 } 18$

Accepted Answer

A)  $0.5286$ 
B)  $0.2872$ 
C)  $1.2552$ 
D)  $1.0791$ 
A)  $0.5286$ 
B)  $0.2872$ 
C)  $1.2552$ 
D)  $1.0791$

Question 2

Provide an appropriate response.
-Use the properties of exponents to write the function of the form $f ( t ) = \mathrm { ka } ^ { t }$, where $\mathrm { k }$ is a constant. $f ( t ) = \left( \frac { 1 } { 2 } \right) ^ { 3 - 4 t }$

Accepted Answer

A)  $f ( t ) = 8 \cdot 16 ^ { t }$ 
B)  $f ( t ) = \left( \frac { 1 } { 16 } \right) 8 ^ { t }$ 
C)  $f ( t ) = \left( \frac { 1 } { 8 } \right) 16 ^ { t }$ 
D)  $f ( t ) = \frac { 1 } { 8 } \left( \frac { 1 } { 16 } \right) ^ { t }$ 
A)  $f ( t ) = 8 \cdot 16 ^ { t }$ 
B)  $f ( t ) = \left( \frac { 1 } { 16 } \right) 8 ^ { t }$ 
C)  $f ( t ) = \left( \frac { 1 } { 8 } \right) 16 ^ { t }$ 
D)  $f ( t ) = \frac { 1 } { 8 } \left( \frac { 1 } { 16 } \right) ^ { t }$

Question 3

Determine whether the statement is true or false.
-If a function f has an inverse, then the graph of $\mathrm { f } ^ { - 1 }$ can be obtained by reflecting the graph of f 
across the x-axis.

Accepted Answer

A) True 
 B)False

Question 4

Find the function value. If the result is irrational, round your answer to the nearest thousandth.
-Let $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$. Find $f ( - 1 )$

Accepted Answer

A)  $- \frac { 1 } { 5 }$ 
B)  $- 5$ 
C)  $\frac { 1 } { 5 }$ 
D)  5 
A)  $- \frac { 1 } { 5 }$ 
B)  $- 5$ 
C)  $\frac { 1 } { 5 }$ 
D)  5

Question 5

Evaluate the logarithm.
-$$\log _ { 5 } \frac { 1 } { 5 }$$

Accepted Answer

A)  5 
B)  0 
C)  1 
D)  $- 1$ 
A)  5 
B)  0 
C)  1 
D)  $- 1$

Question 6

Use the definition of inverses to determine whether f and g are inverses.
-$f ( x ) = 3 x + 9 , \quad g ( x ) = \frac { 1 } { 3 } x - 3$

Accepted Answer

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

Question 7

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent
positive real numbers.
-$$\log _ { b } \sqrt [ 3 ] { \frac { x ^ { 2 } } { y ^ { 4 } z ^ { 7 } } }$$

Accepted Answer

A)  $2 \log _ { b } x - 4 \log _ { b } y - 7 \log _ { b } z$ 
B)  $\frac { 2 } { 3 } \log _ { b } x - \frac { 4 } { 3 } \log _ { b } y - \frac { 7 } { 3 } \log _ { b } z$ 
C)  $\frac { 1 } { 3 } \log _ { b } x ^ { 2 } - \log _ { b } y ^ { 4 } - \log _ { b } z ^ { 7 }$ 
D)  $\frac { 2 } { 3 } \log _ { b } x \div \left( \frac { 4 } { 3 } \log _ { b } y \cdot \frac { 7 } { 3 } \log _ { b } z \right)$ 
A)  $2 \log _ { b } x - 4 \log _ { b } y - 7 \log _ { b } z$ 
B)  $\frac { 2 } { 3 } \log _ { b } x - \frac { 4 } { 3 } \log _ { b } y - \frac { 7 } { 3 } \log _ { b } z$ 
C)  $\frac { 1 } { 3 } \log _ { b } x ^ { 2 } - \log _ { b } y ^ { 4 } - \log _ { b } z ^ { 7 }$ 
D)  $\frac { 2 } { 3 } \log _ { b } x \div \left( \frac { 4 } { 3 } \log _ { b } y \cdot \frac { 7 } { 3 } \log _ { b } z \right)$

Question 8

If the function is one-to-one, find its inverse. If not, write "not one-to-one."
-$$f ( x ) = 2 x ^ { 2 } - 9 , x \geq 0$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \frac { 2 } { \sqrt { x } + 4 }$ 
B)  $f ^ { - 1 } ( x ) = \sqrt { \frac { 2 } { x + 4 } } , x \neq - 9$ 
C)  not a one-to-one 
D)  $f - 1 ( x ) = \sqrt { \frac { x + 9 } { 2 } } , x \geq - 9$ 
A)  $f ^ { - 1 } ( x ) = \frac { 2 } { \sqrt { x } + 4 }$ 
B)  $f ^ { - 1 } ( x ) = \sqrt { \frac { 2 } { x + 4 } } , x \neq - 9$ 
C)  not a one-to-one 
D)  $f - 1 ( x ) = \sqrt { \frac { x + 9 } { 2 } } , x \geq - 9$

Question 9

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth.
-$$3 ^ { 2 x } = 6 ^ { x + 1 }$$

Accepted Answer

A)  $\{ 4.42 \}$ 
B)  $\{ - 2.58 \}$ 
C)  $\{ 2.63 \}$ 
D)  $\{ 1.63 \}$ 
A)  $\{ 4.42 \}$ 
B)  $\{ - 2.58 \}$ 
C)  $\{ 2.63 \}$ 
D)  $\{ 1.63 \}$

Question 10

Find the value. Give an approximation to four decimal places.
-Quail are game birds that fare poorly when their habitat is encroached upon. Wildlife biologists hav discovered that the population $\mathrm { P }$ of quail in a region is related to the percent of the region that has $\mathrm { b } _ { \mathrm { t } }$ paved with roads and parking lots, according to the formula
$$P = k \ln \left( \frac { 32 } { x + 4 } \right) , 0 \leq x \leq 28 \text {, }$$
where $x$ is the percent of the region that has been paved. For a particular rural region, $P = 1400$ wher $x = 0$. Predict what the quail population will be in this region when it becomes $6 \%$ paved. Round your answer to the nearest whole number.

Accepted Answer

A)  836 quail 
B)  0 quail 
C)  783 quail 
D)  794 quail 
A)  836 quail 
B)  0 quail 
C)  783 quail 
D)  794 quail

Question 11

Solve the problem.
-$$\mathrm { y } = \frac { 2 - \log ( 100 - \mathrm { x } ) } { 0.28 }$$ Suppose th can be used to calculate the number of years y for x percent of a population of 298 web-footed sparrows to die. Approximate the percentage (to the nearest whole
Per cent) of web-footed sparrows that died after 3 yr.

Accepted Answer

A)  84% 
B)  80% 
C)  86% 
D)  88% 
A)  84% 
B)  80% 
C)  86% 
D)  88%

Question 12

Choose the one alternative that best completes the statement or answers the question.
-An artifact is discovered at a certain site. If it has $65 \%$ 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)  $5200 \mathrm { yr }$ 
B)  $1497 \mathrm { yr }$ 
C)  $2800 \mathrm { yr }$ 
D)  $3446 \mathrm { yr }$ 
A)  $5200 \mathrm { yr }$ 
B)  $1497 \mathrm { yr }$ 
C)  $2800 \mathrm { yr }$ 
D)  $3446 \mathrm { yr }$

Question 13

Write in logarithmic form.
-$$\left( \frac { 2 } { 7 } \right) ^ { 2 } = \frac { 4 } { 49 }$$

Accepted Answer

A)  $\log _ { 2 / 7 } 2 = \frac { 4 } { 49 }$ 
B)  $\frac { \log _ { 2 } 4 } { \log _ { 2 } 49 } = \frac { 2 } { 7 }$ 
C)  $\log _ { 2 / 7 } \left( \frac { 4 } { 49 } \right) = 2$ 
D)  $\log _ { 2 } \left( \frac { 2 } { 7 } \right) = \frac { 4 } { 49 }$ 
A)  $\log _ { 2 / 7 } 2 = \frac { 4 } { 49 }$ 
B)  $\frac { \log _ { 2 } 4 } { \log _ { 2 } 49 } = \frac { 2 } { 7 }$ 
C)  $\log _ { 2 / 7 } \left( \frac { 4 } { 49 } \right) = 2$ 
D)  $\log _ { 2 } \left( \frac { 2 } { 7 } \right) = \frac { 4 } { 49 }$

Question 14

Choose the one alternative that best completes the statement or answers the question.
-Suppose the government wants to impose a tax on fossil fuels to reduce carbon emissions. The cost benefit is modeled by $\ln ( 1 - \mathrm { P } ) = - 0.0029 - 0.0053 \mathrm { x }$, where $\mathrm { x }$ represents the dollars of tax per ton of carbon emitted and $P$ represents the percent reduction in emissions of carbon. ( $\mathrm { P }$ is in decimal form.) Determine $\mathrm { P }$ when $\mathrm { x } = 56$. Round to three decimal places.

Accepted Answer

A)  $- 0.255$ 
B)  $- 0.15$ 
C)  $1.741$ 
D)  $0.259$ 
A)  $- 0.255$ 
B)  $- 0.15$ 
C)  $1.741$ 
D)  $0.259$

Question 15

Use the graph of f to sketch a graph of the inverse of f using a dashed curve.
-

Accepted Answer

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

Question 16

Determine whether the statement is true or false.
-$$f ( 1 ) = - 3 \text {, then } f ^ { - 1 } ( - 3 ) = 1$$

Accepted Answer

A) True 
 B)False

Question 17

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe
each inverse activity.
-waking up

Accepted Answer

A)  closing the eyes 
B)  falling asleep 
C)  getting up 
D)  going to bed 
A)  closing the eyes 
B)  falling asleep 
C)  getting up 
D)  going to bed

Question 18

Write an equivalent expression in exponential form.
-$$\log _ { 5 } x = 3$$

Accepted Answer

A)  $\{ 8 \}$ 
B)  $\{ 125 \}$ 
C)  $\{ 243 \}$ 
D)  $\{ 15 \}$ 
A)  $\{ 8 \}$ 
B)  $\{ 125 \}$ 
C)  $\{ 243 \}$ 
D)  $\{ 15 \}$

Question 19

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $$a \neq 1 \text { and } b \neq 1$$
-$2 \log _ { 6 } ( 3 x - 3 ) + 6 \log _ { 6 } ( 2 x + 1 )$

Accepted Answer

A)  $\log _ { 6 } \left( ( 3 x - 3 ) ^ { 2 } + ( 2 x + 1 ) ^ { 6 } \right)$ 
B)  $\log _ { 6 } \frac { ( 3 x - 3 ) ^ { 2 } } { ( 2 x + 1 ) ^ { 6 } }$ 
C)  $\log _ { 6 } ( 3 x - 3 ) ^ { 2 } ( 2 x + 1 ) ^ { 6 }$ 
D)  $12 \log _ { 6 } ( 3 x - 3 ) ( 2 x + 1 )$ 
A)  $\log _ { 6 } \left( ( 3 x - 3 ) ^ { 2 } + ( 2 x + 1 ) ^ { 6 } \right)$ 
B)  $\log _ { 6 } \frac { ( 3 x - 3 ) ^ { 2 } } { ( 2 x + 1 ) ^ { 6 } }$ 
C)  $\log _ { 6 } ( 3 x - 3 ) ^ { 2 } ( 2 x + 1 ) ^ { 6 }$ 
D)  $12 \log _ { 6 } ( 3 x - 3 ) ( 2 x + 1 )$

Question 20

Find the function value. If the result is irrational, round your answer to the nearest thousandth.
-Let $f ( x ) = 2 ^ { x }$. Find $f ( 5 )$

Accepted Answer

A)  32 
B)  64 
C)  25 
D)  10 
A)  32 
B)  64 
C)  25 
D)  10

Given $\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$ find the logarithm without using a calculator. - $\log _ { 10 } 18$

Provide an appropriate response. -Use the properties of exponents to write the function of the form $f ( t ) = \mathrm { ka } ^ { t }$ , where $\mathrm { k }$ is a constant. $f ( t ) = \left( \frac { 1 } { 2 } \right) ^ { 3 - 4 t }$

Determine whether the statement is true or false. -If a function f has an inverse, then the graph of $\mathrm { f } ^ { - 1 }$ can be obtained by reflecting the graph of f across the x-axis.

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$ . Find $f ( - 1 )$

Evaluate the logarithm. - $\log _ { 5 } \frac { 1 } { 5 }$

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = 3 x + 9 , \quad g ( x ) = \frac { 1 } { 3 } x - 3$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { b } \sqrt [ 3 ] { \frac { x ^ { 2 } } { y ^ { 4 } z ^ { 7 } } }$

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $f ( x ) = 2 x ^ { 2 } - 9 , x \geq 0$

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $3 ^ { 2 x } = 6 ^ { x + 1 }$

Write in logarithmic form. - $\left( \frac { 2 } { 7 } \right) ^ { 2 } = \frac { 4 } { 49 }$

Use the graph of f to sketch a graph of the inverse of f using a dashed curve. -

Determine whether the statement is true or false. - $f ( 1 ) = - 3 \text {, then } f ^ { - 1 } ( - 3 ) = 1$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe each inverse activity. -waking up

Write an equivalent expression in exponential form. - $\log _ { 5 } x = 3$

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ - $2 \log _ { 6 } ( 3 x - 3 ) + 6 \log _ { 6 } ( 2 x + 1 )$

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $f ( x ) = 2 ^ { x }$ . Find $f ( 5 )$

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

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⁡1018\log _ { 10 } 18log10​18

Provide an appropriate response. -Use the properties of exponents to write the function of the form f(t)=katf ( t ) = \mathrm { ka } ^ { t }f(t)=kat , where k\mathrm { k }k is a constant. f(t)=(12)3−4tf ( t ) = \left( \frac { 1 } { 2 } \right) ^ { 3 - 4 t }f(t)=(21​)3−4t

Determine whether the statement is true or false. -If a function f has an inverse, then the graph of f−1\mathrm { f } ^ { - 1 }f−1 can be obtained by reflecting the graph of f across the x-axis.

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let f(x)=(15)xf ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }f(x)=(51​)x . Find f(−1)f ( - 1 )f(−1)

Evaluate the logarithm. - log⁡515\log _ { 5 } \frac { 1 } { 5 }log5​51​

Use the definition of inverses to determine whether f and g are inverses. - f(x)=3x+9,g(x)=13x−3f ( x ) = 3 x + 9 , \quad g ( x ) = \frac { 1 } { 3 } x - 3f(x)=3x+9,g(x)=31​x−3

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - log⁡bx2y4z73\log _ { b } \sqrt [ 3 ] { \frac { x ^ { 2 } } { y ^ { 4 } z ^ { 7 } } }logb​3y4z7x2​​

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - f(x)=2x2−9,x≥0f ( x ) = 2 x ^ { 2 } - 9 , x \geq 0f(x)=2x2−9,x≥0

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - 32x=6x+13 ^ { 2 x } = 6 ^ { x + 1 }32x=6x+1

Write in logarithmic form. - (27)2=449\left( \frac { 2 } { 7 } \right) ^ { 2 } = \frac { 4 } { 49 }(72​)2=494​