Chapter 10 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

##

Worked example

asymptote

A line which a graph of a function approaches closely but never touches.

common logarithmic function

exponential function

logarithmic function

natural base

e

n

n

natural exponential function

e

natural logarithmic function

one-to-one function

y

x

##

Key Concepts

• Composition of Functions: The composition of functions $f$ and $g,$ is written $f\mathrm{∘}g$ and is defined by

We read $f(g(x))$ as $f$ of $g$ of $x.$

• Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.

• Inverse of a Function Defined by Ordered Pairs: If $f(x)$ is a one-to-one function whose ordered pairs are of the form $(x,y),$ then its inverse function ${f}^{-1}(x)$ is the set of ordered pairs $(y,x).$

• Inverse Functions: For every $x$ in the domain of one-to-one function $f$ and ${f}^{-1},$

• How to Find the Inverse of a One-to-One Function: Step 1. Substitute y for $f(x).$ Step 2. Interchange the variables x and y. Step 3. Solve for y. Step 4. Substitute ${f}^{-1}(x)$ for $y.$ Step 5. Verify that the functions are inverses.

• Properties of the Graph of $f(x)={a}^{x}:$ when $a>1$ when $0<a<1$Domain $(-\infty,\infty)$ Domain $(-\infty,\infty)$ Range $(0,\infty)$ Range $(0,\infty)$$x$-intercept none $x$-intercept none $y$-intercept $(0,1)$$y$-intercept $(0,1)$ Contains $(1,a),\,(-1,\frac{1}{a})$ Contains $(1,a),\,(-1,\frac{1}{a})$ Asymptote $x$-axis, the line $y=0$ Asymptote $x$-axis, the line $y=0$ Basic shape increasing Basic shape decreasing

• One-to-One Property of Exponential Functions: For $a>0$ and $a\ne{}1,$

• How to Solve an Exponential Equation Step 1. Write both sides of the equation with the same base, if possible. Step 2. Write a new equation by setting the exponents equal. Step 3. Solve the equation. Step 4. Check the solution.

• Compound Interest: For a principal, $P,$ invested at an interest rate, $r,$ for $t$ years, the new balance, $A,$ is $\begin{array}{lccccl} A=P{(1+\frac{r}{n})}^{nt} & & & & & \mathrm{when compounded}\,n\,\mathrm{times a year.} \\ A=P{e}^{rt} & & & & & \mathrm{when compounded continuously.} \\ \end{array}$

• Exponential Growth and Decay: For an original amount, ${A}_{0}$ that grows or decays at a rate, $r,$ for a certain time $t,$ the final amount,$A,$ is $A={A}_{0}{e}^{rt}.$

• Properties of the Graph of $y={\mathrm{log}}_{a}x:$ when $a>1$ when $0<a<1$Domain $(0,\infty)$ Domain $(0,\infty)$ Range $(-\infty,\infty)$ Range $(-\infty,\infty)$ x-intercept $(1,0)$ x-intercept $(1,0)$ y-intercept none y-intercept none Contains $(a,1),$$(\frac{1}{a},-1)$ Contains $(a,1),$$(\frac{1}{a},-1)$ Asymptote y-axis Asymptote y-axis Basic shape increasing Basic shape decreasing

• Decibel Level of Sound: The loudness level, $D$, measured in decibels, of a sound of intensity, $I$, measured in watts per square inch is $D=10\mathrm{log}(\frac{I}{{10}^{-12}}).$

• Earthquake Intensity: The magnitude $R$ of an earthquake is measured by $R=\mathrm{log}\,I,$ where $I$ is the intensity of its shock wave.

• Properties of Logarithms

• Inverse Properties of Logarithms For $a>0,$$x>0$ and $a\ne{}1$

• Product Property of Logarithms If $M>0,N>0,\,a>0$ and $a\ne{}1,$ then,

The logarithm of a product is the sum of the logarithms.

• Quotient Property of Logarithms If $M>0,N>0,\,a>0$ and $a\ne{}1,$ then,

The logarithm of a quotient is the difference of the logarithms.

• Power Property of Logarithms If $M>0,\,a>0,\,a\ne{}1$ and $p$ is any real number then,

The log of a number raised to a power is the product of the power times the log of the number.

• Properties of Logarithms Summary If $M>0,\,a>0,\,a\ne{}1$ and $p$ is any real number then, Property Base $a$ Base $e$${\mathrm{log}}_{a}1=0$$\mathrm{ln}1=0$${\mathrm{log}}_{a}a=1$$\mathrm{ln}\,e=1$ Inverse Properties $\begin{array}{l} \,{a}^{{\mathrm{log}}_{a}x}=x \\ \,{\mathrm{log}}_{a}{a}^{x}=x \\ \end{array}$$\begin{array}{l} \,{e}^{\mathrm{ln}\,x}=x \\ \,\mathrm{ln}\,{e}^{x}=x \\ \end{array}$ Product Property of Logarithms ${\mathrm{log}}_{a}(M\cdot{}N)={\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N$$\mathrm{ln}(M\,\cdot\,N)=\mathrm{ln}\,M+\mathrm{ln}\,N$ Quotient Property of Logarithms $\,{\mathrm{log}}_{a}\frac{M}{N}={\mathrm{log}}_{a}M-{\mathrm{log}}_{a}N$$\,\mathrm{ln}\frac{M}{N}=\mathrm{ln}\,M-\mathrm{ln}\,N$ Power Property of Logarithms $\,{\mathrm{log}}_{a}{M}^{p}=p\,{\mathrm{log}}_{a}M$$\,\mathrm{ln}\,{M}^{p}=p\mathrm{ln}\,M$

• Change-of-Base Formula For any logarithmic bases a and b, and $M>0,$

• One-to-One Property of Logarithmic Equations: For $M>0,N>0,\,a\,>\,0,$ and $a\ne{}1$ is any real number:

• Compound Interest: For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:

• Exponential Growth and Decay: For an original amount, ${A}_{0}$ that grows or decays at a rate, r, for a certain time t, the final amount, A, is $A={A}_{0}{e}^{rt}.$

##

Review Exercises

Find and Evaluate Composite Functions

In the following exercises, for each pair of functions, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

In the following exercises, evaluate the composition.

Determine Whether a Function is One-to-One

In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

Find the Inverse of a Function

In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.

In the following exercise, graph the inverse of the one-to-one function shown.

In the following exercises, verify that the functions are inverse functions.

In the following exercises, find the inverse of each function.

Graph Exponential Functions

In the following exercises, graph each of the following functions.

Solve Exponential Equations

In the following exercises, solve each equation.

Use Exponential Models in Applications

In the following exercises, solve.

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

In the following exercises, convert each logarithmic equation to exponential form.

Evaluate Logarithmic Functions

In the following exercises, solve for x.

In the following exercises, find the exact value of each logarithm without using a calculator.

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

Use Logarithmic Models in Applications

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

Use the Change-of-Base Formula

In the following exercises, rounding to three decimal places, approximate each logarithm.

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

Use Exponential Models in Applications

##

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.

In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

Key Terms

asymptote

A line which a graph of a function approaches closely but never touches.

common logarithmic function

exponential function

logarithmic function

natural base

e

n

n

natural exponential function

e

natural logarithmic function

one-to-one function

y

x

##

Key Concepts

• Composition of Functions: The composition of functions $f$ and $g,$ is written $f\mathrm{∘}g$ and is defined by

We read $f(g(x))$ as $f$ of $g$ of $x.$

• Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.

• Inverse of a Function Defined by Ordered Pairs: If $f(x)$ is a one-to-one function whose ordered pairs are of the form $(x,y),$ then its inverse function ${f}^{-1}(x)$ is the set of ordered pairs $(y,x).$

• Inverse Functions: For every $x$ in the domain of one-to-one function $f$ and ${f}^{-1},$

• How to Find the Inverse of a One-to-One Function: Step 1. Substitute y for $f(x).$ Step 2. Interchange the variables x and y. Step 3. Solve for y. Step 4. Substitute ${f}^{-1}(x)$ for $y.$ Step 5. Verify that the functions are inverses.

• Properties of the Graph of $f(x)={a}^{x}:$ when $a>1$ when $0<a<1$Domain $(-\infty,\infty)$ Domain $(-\infty,\infty)$ Range $(0,\infty)$ Range $(0,\infty)$$x$-intercept none $x$-intercept none $y$-intercept $(0,1)$$y$-intercept $(0,1)$ Contains $(1,a),\,(-1,\frac{1}{a})$ Contains $(1,a),\,(-1,\frac{1}{a})$ Asymptote $x$-axis, the line $y=0$ Asymptote $x$-axis, the line $y=0$ Basic shape increasing Basic shape decreasing

• One-to-One Property of Exponential Functions: For $a>0$ and $a\ne{}1,$

• How to Solve an Exponential Equation Step 1. Write both sides of the equation with the same base, if possible. Step 2. Write a new equation by setting the exponents equal. Step 3. Solve the equation. Step 4. Check the solution.

• Compound Interest: For a principal, $P,$ invested at an interest rate, $r,$ for $t$ years, the new balance, $A,$ is $\begin{array}{lccccl} A=P{(1+\frac{r}{n})}^{nt} & & & & & \mathrm{when compounded}\,n\,\mathrm{times a year.} \\ A=P{e}^{rt} & & & & & \mathrm{when compounded continuously.} \\ \end{array}$

• Exponential Growth and Decay: For an original amount, ${A}_{0}$ that grows or decays at a rate, $r,$ for a certain time $t,$ the final amount,$A,$ is $A={A}_{0}{e}^{rt}.$

• Properties of the Graph of $y={\mathrm{log}}_{a}x:$ when $a>1$ when $0<a<1$Domain $(0,\infty)$ Domain $(0,\infty)$ Range $(-\infty,\infty)$ Range $(-\infty,\infty)$ x-intercept $(1,0)$ x-intercept $(1,0)$ y-intercept none y-intercept none Contains $(a,1),$$(\frac{1}{a},-1)$ Contains $(a,1),$$(\frac{1}{a},-1)$ Asymptote y-axis Asymptote y-axis Basic shape increasing Basic shape decreasing

• Decibel Level of Sound: The loudness level, $D$, measured in decibels, of a sound of intensity, $I$, measured in watts per square inch is $D=10\mathrm{log}(\frac{I}{{10}^{-12}}).$

• Earthquake Intensity: The magnitude $R$ of an earthquake is measured by $R=\mathrm{log}\,I,$ where $I$ is the intensity of its shock wave.

• Properties of Logarithms

• Inverse Properties of Logarithms For $a>0,$$x>0$ and $a\ne{}1$

• Product Property of Logarithms If $M>0,N>0,\,a>0$ and $a\ne{}1,$ then,

The logarithm of a product is the sum of the logarithms.

• Quotient Property of Logarithms If $M>0,N>0,\,a>0$ and $a\ne{}1,$ then,

The logarithm of a quotient is the difference of the logarithms.

• Power Property of Logarithms If $M>0,\,a>0,\,a\ne{}1$ and $p$ is any real number then,

The log of a number raised to a power is the product of the power times the log of the number.

• Properties of Logarithms Summary If $M>0,\,a>0,\,a\ne{}1$ and $p$ is any real number then, Property Base $a$ Base $e$${\mathrm{log}}_{a}1=0$$\mathrm{ln}1=0$${\mathrm{log}}_{a}a=1$$\mathrm{ln}\,e=1$ Inverse Properties $\begin{array}{l} \,{a}^{{\mathrm{log}}_{a}x}=x \\ \,{\mathrm{log}}_{a}{a}^{x}=x \\ \end{array}$$\begin{array}{l} \,{e}^{\mathrm{ln}\,x}=x \\ \,\mathrm{ln}\,{e}^{x}=x \\ \end{array}$ Product Property of Logarithms ${\mathrm{log}}_{a}(M\cdot{}N)={\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N$$\mathrm{ln}(M\,\cdot\,N)=\mathrm{ln}\,M+\mathrm{ln}\,N$ Quotient Property of Logarithms $\,{\mathrm{log}}_{a}\frac{M}{N}={\mathrm{log}}_{a}M-{\mathrm{log}}_{a}N$$\,\mathrm{ln}\frac{M}{N}=\mathrm{ln}\,M-\mathrm{ln}\,N$ Power Property of Logarithms $\,{\mathrm{log}}_{a}{M}^{p}=p\,{\mathrm{log}}_{a}M$$\,\mathrm{ln}\,{M}^{p}=p\mathrm{ln}\,M$

• Change-of-Base Formula For any logarithmic bases a and b, and $M>0,$

• One-to-One Property of Logarithmic Equations: For $M>0,N>0,\,a\,>\,0,$ and $a\ne{}1$ is any real number:

• Compound Interest: For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:

• Exponential Growth and Decay: For an original amount, ${A}_{0}$ that grows or decays at a rate, r, for a certain time t, the final amount, A, is $A={A}_{0}{e}^{rt}.$

##

Review Exercises

Find and Evaluate Composite Functions

In the following exercises, for each pair of functions, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

In the following exercises, evaluate the composition.

Determine Whether a Function is One-to-One

In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

Find the Inverse of a Function

In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.

In the following exercise, graph the inverse of the one-to-one function shown.

In the following exercises, verify that the functions are inverse functions.

In the following exercises, find the inverse of each function.

Graph Exponential Functions

In the following exercises, graph each of the following functions.

Solve Exponential Equations

In the following exercises, solve each equation.

Use Exponential Models in Applications

In the following exercises, solve.

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

In the following exercises, convert each logarithmic equation to exponential form.

Evaluate Logarithmic Functions

In the following exercises, solve for x.

In the following exercises, find the exact value of each logarithm without using a calculator.

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

Use Logarithmic Models in Applications

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

Use the Change-of-Base Formula

In the following exercises, rounding to three decimal places, approximate each logarithm.

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

Use Exponential Models in Applications

##

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.

In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

1. $f(x)=4x$ and$g(x)={x}^{2}+3x$

2. For functions$f(x)=2{x}^{3}+5$ and$g(x)=3{x}^{2}-7,$ findⓐ $(f\mathrm{∘}g)(-1)$ⓑ $(g\mathrm{∘}f)(-2)$ⓒ $(g\mathrm{∘}g)(1)$

3. $\mathrm{\{}(-3,0),(-2,-2),(-1,0),(0,1),$$(1,2),(2,1),(3,-1)\mathrm{\}}$

4. ⓐ

ⓑ

1. $\mathrm{\{}(-3,10),(-2,5),(-1,2),(0,1)\mathrm{\}}$

2. $f(x)=3x+7$ and$g(x)=\frac{x-7}{3}$

3. $f(x)=6x-11$

4. $f(x)=\frac{1}{x+5}$

1. $f(x)={4}^{x}$

2. $g(x)={(0.75)}^{x}$

3. $f(x)={(2.3)}^{x}-3$

4. $f(x)=-{e}^{x}$

1. ${2}^{{x}^{2}}=16$

2. ${5}^{{x}^{2}+2x}=\frac{1}{5}$

3. $\frac{{e}^{{x}^{2}}}{{e}^{15}}={e}^{2x}$

4. Sayed deposits $20,000 in an investment account. What will be the value of his investment in 30 years if the investment is earning $7\%$ per year and is compounded continuously?

1. In the last five years the population of the United States has grown at a rate of $0.7\%$ per year to about 318,900,000. If this rate continues, what will be the population in 5 more years?

2. ${10}^{-3}=\frac{1}{1,000}$

3. ${e}^{y}=16$

4. $5=\mathrm{log}100,000$

1. ${\mathrm{log}}_{x}125=3$

2. ${\mathrm{log}}_{\frac{1}{2}}\frac{1}{16}=x$

3. $y={\mathrm{log}}_{5}x$

4. $y={\mathrm{log}}_{0.8}x$

1. $\mathrm{ln}\,x=-3$

2. $\mathrm{ln}\,{e}^{3x}=24$

3. What is the decibel level of a train whistle with intensity ${10}^{-3}$ watts per square inch?

4. ⓐ ${5}^{{\mathrm{log}}_{5}13}$ ⓑ ${\mathrm{log}}_{3}{3}^{-9}$

1. ⓐ ${e}^{\mathrm{ln}8}$ ⓑ $\mathrm{ln}\,{e}^{5}$

2. $\mathrm{log}10,000m$

3. $\mathrm{ln}\frac{{e}^{5}}{2}$

4. ${\mathrm{log}}_{4}\sqrt[7]{z}$

1. ${\mathrm{log}}_{5}\frac{8{a}^{2}{b}^{6}c}{{d}^{3}}$

2. ${\mathrm{log}}_{6}\sqrt[3]{\frac{7{x}^{2}}{6{y}^{3}{z}^{5}}}$

3. $3{\mathrm{log}}_{3}x+7{\mathrm{log}}_{3}y$

4. $\frac{1}{4}\mathrm{log}y-2\mathrm{log}(y-3)$

1. ${\mathrm{log}}_{2}x+{\mathrm{log}}_{2}(x-2)=3$

2. ${\mathrm{log}}_{4}(x-2)+{\mathrm{log}}_{4}(x+5)={\mathrm{log}}_{4}8$

3. ${2}^{x}=101$

4. ${(\frac{1}{3})}^{x}=7$

1. ${e}^{x-4}+8=23$

2. Elise invests $4500 in an account that compounds interest monthly and earns $6\%.$ How long will it take for her money to double?

3. Mouse populations can double in 8 months $(A=2{A}_{0}).$ How long will it take for a mouse population to triple?

4. For the functions, $f(x)=6x+1$ and $g(x)=8x-3,$ find ⓐ $(f\mathrm{∘}g)(x),$ ⓑ $(g\mathrm{∘}f)(x),$ and ⓒ $(f\cdot{}g)(x).$

1. Determine whether each graph is the graph of a function and if so, is it one-to-one.

ⓐ

ⓑ

2. Find the inverse of the function $f(x)={x}^{5}-9.$

3. Solve the equation ${2}^{2x-4}=64.$

4. Megan invested $21,000 in a savings account. If the interest rate is $5\%,$ how much will be in the account in 8 years by each method of compounding?ⓐ compound quarterlyⓑ compound monthlyⓒ compound continuously.

1. Convert the equation from logarithmic equation to exponential form: $3={\mathrm{log}}_{7}343$

2. What is the decibel level of a small fan with intensity ${10}^{-8}$ watts per square inch?

3. ${\mathrm{log}}_{5}25ab$

4. ${\mathrm{log}}_{2}\sqrt[4]{\frac{5{x}^{3}}{16{y}^{2}{z}^{7}}}$

1. $\frac{1}{6}\mathrm{log}\,x-3\mathrm{log}(x+5)$

2. Solve for x:${\mathrm{log}}_{7}(x+2)+{\mathrm{log}}_{7}(x-3)={\mathrm{log}}_{7}24$

3. $5{e}^{x-4}=40$

4. Researchers recorded that a certain bacteria population grew from 500 to 700 in 5 hours. At this rate of growth, how many bacteria will there be in 20 hours?