Skip to Content
OpenStax Logo
Intermediate Algebra 2e

10.3 Evaluate and Graph Logarithmic Functions

Intermediate Algebra 2e10.3 Evaluate and Graph Logarithmic Functions
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:

  • Convert between exponential and logarithmic form
  • Evaluate logarithmic functions
  • Graph Logarithmic functions
  • Solve logarithmic equations
  • Use logarithmic models in applications
Be Prepared 10.7

Before you get started, take this readiness quiz.

Solve: x2=81.x2=81.
If you missed this problem, review Example 6.46.

Be Prepared 10.8

Evaluate: 3−2.3−2.
If you missed this problem, review Example 5.15.

Be Prepared 10.9

Solve: 24=3x5.24=3x5.
If you missed this problem, review Example 2.2.

We have spent some time finding the inverse of many functions. It works well to ‘undo’ an operation with another operation. Subtracting ‘undoes’ addition, multiplication ‘undoes’ division, taking the square root ‘undoes’ squaring.

As we studied the exponential function, we saw that it is one-to-one as its graphs pass the horizontal line test. This means an exponential function does have an inverse. If we try our algebraic method for finding an inverse, we run into a problem.

Rewrite withy=f(x).Interchange the variablesxandy.f(x)=axy=axx=aySolve fory.Oops! We have no way to solve fory!Rewrite withy=f(x).Interchange the variablesxandy.f(x)=axy=axx=aySolve fory.Oops! We have no way to solve fory!

To deal with this we define the logarithm function with base a to be the inverse of the exponential function f(x)=ax.f(x)=ax. We use the notation f−1(x)=logaxf−1(x)=logax and say the inverse function of the exponential function is the logarithmic function.

Logarithmic Function

The function f(x)=logaxf(x)=logax is the logarithmic function with base aa, where a>0,a>0,x>0,x>0, and a1.a1.

y=logaxis equivalent tox=ayy=logaxis equivalent tox=ay

Convert Between Exponential and Logarithmic Form

Since the equations y=logaxy=logax and x=ayx=ay are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let’s take a close look at the equations. See Figure 10.3. Notice the positions of the exponent and base.

This figure shows the expression y equals log sub a of x, where y is the exponent and a is the base. Next to this expression we have x equals a to the y, where again y is the exponent and a is the base.
Figure 10.3

If we realize the logarithm is the exponent it makes the conversion easier. You may want to repeat, “base to the exponent give us the number.”

Example 10.18

Convert to logarithmic form: 23=8,23=8, 512=5,512=5, and (12)x=116.(12)x=116.

Try It 10.35

Convert to logarithmic form: 32=932=9 712=7712=7 (13)x=127(13)x=127

Try It 10.36

Convert to logarithmic form: 43=6443=64 413=43413=43 (12)x=132(12)x=132

In the next example we do the reverse—convert logarithmic form to exponential form.

Example 10.19

Convert to exponential form: 2=log864,2=log864, 0=log41,0=log41, and 3=log1011000.3=log1011000.

Try It 10.37

Convert to exponential form: 3=log4643=log464 0=logx10=logx1 −2=log101100−2=log101100

Try It 10.38

Convert to exponential form: 3=log3273=log327 0=logx10=logx1 −1=log10110−1=log10110

Evaluate Logarithmic Functions

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation.

Example 10.20

Find the value of x: logx36=2,logx36=2, log4x=3,log4x=3, and log1218=x.log1218=x.

Try It 10.39

Find the value of x:x: logx64=2logx64=2 log5x=3log5x=3 log1214=xlog1214=x

Try It 10.40

Find the value of x:x: logx81=2logx81=2 log3x=5log3x=5 log13127=xlog13127=x

When see an expression such as log327,log327, we can find its exact value two ways. By inspection we realize it means 33 to what power will be 27?27? Since 33=27,33=27, we know log327=3.log327=3. An alternate way is to set the expression equal to xx and then convert it into an exponential equation.

Example 10.21

Find the exact value of each logarithm without using a calculator: log525,log525, log93,log93, and log2116.log2116.

Try It 10.41

Find the exact value of each logarithm without using a calculator: log12144log12144 log42log42 log2132log2132

Try It 10.42

Find the exact value of each logarithm without using a calculator: log981log981 log82log82 log319log319

Graph Logarithmic Functions

To graph a logarithmic function y=logax,y=logax, it is easiest to convert the equation to its exponential form, x=ay.x=ay. Generally, when we look for ordered pairs for the graph of a function, we usually choose an x-value and then determine its corresponding y-value. In this case you may find it easier to choose y-values and then determine its corresponding x-value.

Example 10.22

Graph y=log2x.y=log2x.

Try It 10.43

Graph: y=log3x.y=log3x.

Try It 10.44

Graph: y=log5x.y=log5x.

The graphs of y=log2x,y=log2x,y=log3x,y=log3x, and y=log5xy=log5x are the shape we expect from a logarithmic function where a>1.a>1.

We notice that for each function the graph contains the point (1,0).(1,0). This make sense because 0=loga10=loga1 means a0=1a0=1 which is true for any a.

The graph of each function, also contains the point (a,1).(a,1). This makes sense as 1=logaa1=logaa means a1=a.a1=a. which is true for any a.

Notice too, the graph of each function y=logaxy=logax also contains the point (1a,−1).(1a,−1). This makes sense as −1=loga1a−1=loga1a means a−1=1a,a−1=1a, which is true for any a.

Look at each graph again. Now we will see that many characteristics of the logarithm function are simply ’mirror images’ of the characteristics of the corresponding exponential function.

What is the domain of the function? The graph never hits the y-axis. The domain is all positive numbers. We write the domain in interval notation as (0,).(0,).

What is the range for each function? From the graphs we can see that the range is the set of all real numbers. There is no restriction on the range. We write the range in interval notation as (,).(,).

When the graph approaches the y-axis so very closely but will never cross it, we call the line x=0,x=0, the y-axis, a vertical asymptote.

Properties of the Graph of y=logaxy=logax when a>1a>1

Domain (0,)(0,)
Range (,)(,)
x-interceptx-intercept (1,0)(1,0)
y-intercepty-intercept None
Contains (a,1),(a,1),(1a,−1)(1a,−1)
Asymptote y-axisy-axis
This figure shows the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1).

Our next example looks at the graph of y=logaxy=logax when 0<a<1.0<a<1.

Example 10.23

Graph y=log13x.y=log13x.

Try It 10.45

Graph: y=log12x.y=log12x.

Try It 10.46

Graph: y=log14x.y=log14x.

Now, let’s look at the graphs y=log12x,y=log13xy=log12x,y=log13x and y=log14xy=log14x, so we can identify some of the properties of logarithmic functions where 0<a<1.0<a<1.

The graphs of all have the same basic shape. While this is the shape we expect from a logarithmic function where 0<a<1.0<a<1.

We notice, that for each function again, the graph contains the points,(1,0),(1,0),(a,1),(a,1),(1a,−1).(1a,−1). This make sense for the same reasons we argued above.

We notice the domain and range are also the same—the domain is (0,)(0,) and the range is (,).(,). The yy-axis is again the vertical asymptote.

We will summarize these properties in the chart below. Which also include when a>1.a>1.

Properties of the Graph of y=logaxy=logax

when a>1a>1 when 0<a<10<a<1
Domain (0,)(0,) Domain (0,)(0,)
Range (,)(,) Range (,)(,)
xx-intercept (1,0)(1,0) xx-intercept (1,0)(1,0)
yy-intercept none yy-intercept None
Contains (a,1),(a,1),(1a,−1)(1a,−1) Contains (a,1),(a,1),(1a,−1)(1a,−1)
Asymptote yy-axis Asymptote yy-axis
Basic shape increasing Basic shape Decreasing
This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).

We talked earlier about how the logarithmic function f−1(x)=logaxf−1(x)=logax is the inverse of the exponential function f(x)=ax.f(x)=ax. The graphs in Figure 10.4 show both the exponential (blue) and logarithmic (red) functions on the same graph for both a>1a>1 and 0<a<1.0<a<1.

This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). It also shows the exponential curve going through the points (1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line. This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1). It also shows the exponential curve going through the points (negative 1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line.
Figure 10.4

Notice how the graphs are reflections of each other through the line y=x.y=x. We know this is true of inverse functions. Keeping a visual in your mind of these graphs will help you remember the domain and range of each function. Notice the x-axis is the horizontal asymptote for the exponential functions and the y-axis is the vertical asymptote for the logarithmic functions.

Solve Logarithmic Equations

When we talked about exponential functions, we introduced the number e. Just as e was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base e is called the natural logarithmic function. The function f(x)=logexf(x)=logex is generally written f(x)=lnxf(x)=lnx and we read it as “el en of x.x.

Natural Logarithmic Function

The function f(x)=lnxf(x)=lnx is the natural logarithmic function with base e,e, where x>0.x>0.

y=lnxis equivalent tox=eyy=lnxis equivalent tox=ey

When the base of the logarithm function is 10, we call it the common logarithmic function and the base is not shown. If the base a of a logarithm is not shown, we assume it is 10.

Common Logarithmic Function

The function f(x)=logxf(x)=logx is the common logarithmic function with base1010, where x>0.x>0.

y=logxis equivalent tox=10yy=logxis equivalent tox=10y
It will be important for you to use your calculator to evaluate both common and natural logarithms. Find the log and ln keys on your calculator.

To solve logarithmic equations, one strategy is to change the equation to exponential form and then solve the exponential equation as we did before. As we solve logarithmic equations, y=logaxy=logax, we need to remember that for the base a, a>0a>0 and a1.a1. Also, the domain is x>0.x>0. Just as with radical equations, we must check our solutions to eliminate any extraneous solutions.

Example 10.24

Solve: loga49=2loga49=2 and lnx=3.lnx=3.

Try It 10.47

Solve: loga121=2loga121=2 lnx=7lnx=7

Try It 10.48

Solve: loga64=3loga64=3 lnx=9lnx=9

Example 10.25

Solve: log2(3x5)=4log2(3x5)=4 and lne2x=4.lne2x=4.

Try It 10.49

Solve: log2(5x1)=6log2(5x1)=6 lne3x=6lne3x=6

Try It 10.50

Solve: log3(4x+3)=3log3(4x+3)=3 lne4x=4lne4x=4

Use Logarithmic Models in Applications

There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The 10−1210−12 in the formula represents the intensity of sound that is barely audible.

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=10log(I10−12)D=10log(I10−12)

Example 10.26

Extended exposure to noise that measures 85 dB can cause permanent damage to the inner ear which will result in hearing loss. What is the decibel level of music coming through ear phones with intensity 10−210−2 watts per square inch?

Try It 10.51

What is the decibel level of one of the new quiet dishwashers with intensity 10−710−7 watts per square inch?

Try It 10.52

What is the decibel level heavy city traffic with intensity 10−310−3 watts per square inch?

The magnitude RR of an earthquake is measured by a logarithmic scale called the Richter scale. The model is R=logI,R=logI, where II is the intensity of the shock wave. This model provides a way to measure earthquake intensity.

Earthquake Intensity

The magnitude R of an earthquake is measured by R=logI,R=logI, where I is the intensity of its shock wave.

Example 10.27

In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. Over 80% of the city was destroyed by the resulting fires. In 2014, Los Angeles experienced a moderate earthquake that measured 5.1 on the Richter scale and caused $108 million dollars of damage. Compare the intensities of the two earthquakes.

Try It 10.53

In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. In 1989, the Loma Prieta earthquake also affected the San Francisco area, and measured 6.9 on the Richter scale. Compare the intensities of the two earthquakes.

Try It 10.54

In 2014, Chile experienced an intense earthquake with a magnitude of 8.2 on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured 5.1 on the Richter scale. Compare the intensities of the two earthquakes.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with evaluating and graphing logarithmic functions.

Section 10.3 Exercises

Practice Makes Perfect

Convert Between Exponential and Logarithmic Form

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

126.

42=1642=16

127.

25=3225=32

128.

33=2733=27

129.

53=12553=125

130.

103=1000103=1000

131.

10−2=110010−2=1100

132.

x12=3x12=3

133.

x13=63x13=63

134.

32x=32432x=324

135.

17x=17517x=175

136.

(14)2=116(14)2=116

137.

(13)4=181(13)4=181

138.

3−2=193−2=19

139.

4−3=1644−3=164

140.

ex=6ex=6

141.

e3=xe3=x

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

142.

3=log4643=log464

143.

6=log2646=log264

144.

4=logx814=logx81

145.

5=logx325=logx32

146.

0=log1210=log121

147.

0=log710=log71

148.

1=log331=log33

149.

1=log991=log99

150.

−4=log10110,000−4=log10110,000

151.

3=log101,0003=log101,000

152.

5=logex5=logex

153.

x=loge43x=loge43

Evaluate Logarithmic Functions

In the following exercises, find the value of xx in each logarithmic equation.

154.

logx49=2logx49=2

155.

logx121=2logx121=2

156.

logx27=3logx27=3

157.

logx64=3logx64=3

158.

log3x=4log3x=4

159.

log5x=3log5x=3

160.

log2x=−6log2x=−6

161.

log3x=−5log3x=−5

162.

log14116=xlog14116=x

163.

log1319=xlog1319=x

164.

log1464=xlog1464=x

165.

log1981=xlog1981=x

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

166.

log749log749

167.

log636log636

168.

log41log41

169.

log51log51

170.

log164log164

171.

log273log273

172.

log122log122

173.

log124log124

174.

log2116log2116

175.

log3127log3127

176.

log4116log4116

177.

log9181log9181

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

178.

y=log2xy=log2x

179.

y=log4xy=log4x

180.

y=log6xy=log6x

181.

y=log7xy=log7x

182.

y=log1.5xy=log1.5x

183.

y=log2.5xy=log2.5x

184.

y=log13xy=log13x

185.

y=log15xy=log15x

186.

y=log0.4xy=log0.4x

187.

y=log0.6xy=log0.6x

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

188.

loga16=2loga16=2

189.

loga81=2loga81=2

190.

loga8=3loga8=3

191.

loga27=3loga27=3

192.

loga32=2loga32=2

193.

loga24=3loga24=3

194.

lnx=5lnx=5

195.

lnx=4lnx=4

196.

log2(5x+1)=4log2(5x+1)=4

197.

log2(6x+2)=5log2(6x+2)=5

198.

log3(4x3)=2log3(4x3)=2

199.

log3(5x4)=4log3(5x4)=4

200.

log4(5x+6)=3log4(5x+6)=3

201.

log4(3x2)=2log4(3x2)=2

202.

lne4x=8lne4x=8

203.

lne2x=6lne2x=6

204.

logx2=2logx2=2

205.

log(x225)=2log(x225)=2

206.

log2(x24)=5log2(x24)=5

207.

log3(x2+2)=3log3(x2+2)=3

Use Logarithmic Models in Applications

In the following exercises, use a logarithmic model to solve.

208.

What is the decibel level of normal conversation with intensity 10−610−6 watts per square inch?

209.

What is the decibel level of a whisper with intensity 10−1010−10 watts per square inch?

210.

What is the decibel level of the noise from a motorcycle with intensity 10−210−2 watts per square inch?

211.

What is the decibel level of the sound of a garbage disposal with intensity 10−210−2 watts per square inch?

212.

In 2014, Chile experienced an intense earthquake with a magnitude of 8.28.2 on the Richter scale. In 2010, Haiti also experienced an intense earthquake which measured 7.07.0 on the Richter scale. Compare the intensities of the two earthquakes.

213.

The Los Angeles area experiences many earthquakes. In 1994, the Northridge earthquake measured magnitude of 6.76.7 on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured 5.15.1 on the Richter scale. Compare the intensities of the two earthquakes.

Writing Exercises

214.

Explain how to change an equation from logarithmic form to exponential form.

215.

Explain the difference between common logarithms and natural logarithms.

216.

Explain why logaax=x.logaax=x.

217.

Explain how to find the log732log732 on your calculator.

Self Check


After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four rows and five columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads Convert between exponential and logarithmic form, evaluate logarithmic functions, graph logarithmic functions, solve logarithmic equations, and use logarithmic models in applications. The rest of the cells are blank.

After reviewing this checklist, what will you do to become confident for all objectives?

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
Citation information

© Sep 2, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.