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Intermediate Algebra 2e

10.4 Use the Properties of Logarithms

Intermediate Algebra 2e10.4 Use the Properties of Logarithms
  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:
  • Use the properties of logarithms
  • Use the Change of Base Formula
Be Prepared 10.10

Before you get started, take this readiness quiz.

Evaluate: a0a0 a1.a1.
If you missed this problem, review Example 5.14.

Be Prepared 10.11

Write with a rational exponent: x2y3.x2y3.
If you missed this problem, review Example 8.27.

Be Prepared 10.12

Round to three decimal places: 2.5646415.
If you missed this problem, review Example 1.34.

Use the Properties of Logarithms

Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.

The first two properties derive from the definition of logarithms. Since a0=1,a0=1, we can convert this to logarithmic form and get loga1=0.loga1=0. Also, since a1=a,a1=a, we get logaa=1.logaa=1.

Properties of Logarithms

loga1=0logaa=1loga1=0logaa=1

In the next example we could evaluate the logarithm by converting to exponential form, as we have done previously, but recognizing and then applying the properties saves time.

Example 10.28

Evaluate using the properties of logarithms: log81log81 and log66.log66.

Try It 10.55

Evaluate using the properties of logarithms: log131log131 log99.log99.

Try It 10.56

Evaluate using the properties of logarithms: log51log51 log77.log77.

The next two properties can also be verified by converting them from exponential form to logarithmic form, or the reverse.

The exponential equation alogax=xalogax=x converts to the logarithmic equation logax=logax,logax=logax, which is a true statement for positive values for x only.

The logarithmic equation logaax=xlogaax=x converts to the exponential equation ax=ax,ax=ax, which is also a true statement.

These two properties are called inverse properties because, when we have the same base, raising to a power “undoes” the log and taking the log “undoes” raising to a power. These two properties show the composition of functions. Both ended up with the identity function which shows again that the exponential and logarithmic functions are inverse functions.

Inverse Properties of Logarithms

For a>0,a>0,x>0x>0 and a1,a1,

alogax=xlogaax=xalogax=xlogaax=x

In the next example, apply the inverse properties of logarithms.

Example 10.29

Evaluate using the properties of logarithms: 4log494log49 and log335.log335.

Try It 10.57

Evaluate using the properties of logarithms: 5log5155log515 log774.log774.

Try It 10.58

Evaluate using the properties of logarithms: 2log282log28 log2215.log2215.

There are three more properties of logarithms that will be useful in our work. We know exponential functions and logarithmic function are very interrelated. Our definition of logarithm shows us that a logarithm is the exponent of the equivalent exponential. The properties of exponents have related properties for exponents.

In the Product Property of Exponents, am·an=am+n,am·an=am+n, we see that to multiply the same base, we add the exponents. The Product Property of Logarithms, logaM·N=logaM+logaNlogaM·N=logaM+logaN tells us to take the log of a product, we add the log of the factors.

Product Property of Logarithms

If M>0,N>0,a>0M>0,N>0,a>0 and a1,a1, then,

loga(M·N)=logaM+logaNloga(M·N)=logaM+logaN

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

We use this property to write the log of a product as a sum of the logs of each factor.

Example 10.30

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible: log37xlog37x and log464xy.log464xy.

Try It 10.59

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

log33xlog33x log28xylog28xy

Try It 10.60

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

log99xlog99x log327xylog327xy

Similarly, in the Quotient Property of Exponents, aman=amn,aman=amn, we see that to divide the same base, we subtract the exponents. The Quotient Property of Logarithms, logaMN=logaMlogaNlogaMN=logaMlogaN tells us to take the log of a quotient, we subtract the log of the numerator and denominator.

Quotient Property of Logarithms

If M>0,N>0,a>0M>0,N>0,a>0 and a1,a1, then,

logaMN=logaMlogaNlogaMN=logaMlogaN

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

Note that logaMlogaNloga(MN).logaMlogaNloga(MN).

We use this property to write the log of a quotient as a difference of the logs of each factor.

Example 10.31

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.
log557log557 and logx100logx100

Try It 10.61

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

log434log434 logx1000logx1000

Try It 10.62

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

log254log254 log10ylog10y

The third property of logarithms is related to the Power Property of Exponents, (am)n=am·n,(am)n=am·n, we see that to raise a power to a power, we multiply the exponents. The Power Property of Logarithms, logaMp=plogaMlogaMp=plogaM tells us to take the log of a number raised to a power, we multiply the power times the log of the number.

Power Property of Logarithms

If M>0,a>0,a1M>0,a>0,a1 and pp is any real number then,

logaMp=plogaMlogaMp=plogaM

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

We use this property to write the log of a number raised to a power as the product of the power times the log of the number. We essentially take the exponent and throw it in front of the logarithm.

Example 10.32

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.
log543log543 and logx10logx10

Try It 10.63

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

log754log754 logx100logx100

Try It 10.64

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

log237log237 logx20logx20

We summarize the Properties of Logarithms here for easy reference. While the natural logarithms are a special case of these properties, it is often helpful to also show the natural logarithm version of each property.

Properties of Logarithms

If M>0,N>0,a>0,a1M>0,N>0,a>0,a1 and pp is any real number then,

Property Base aa Base ee
loga1=0loga1=0 ln1=0ln1=0
logaa=1logaa=1 lne=1lne=1
Inverse Properties alogax=xlogaax=xalogax=xlogaax=x elnx=x lnex=xelnx=x lnex=x
Product Property of Logarithms loga(M·N)=logaM+logaNloga(M·N)=logaM+logaN ln(M·N)=lnM+lnNln(M·N)=lnM+lnN
Quotient Property of Logarithms logaMN=logaMlogaNlogaMN=logaMlogaN lnMN=lnMlnNlnMN=lnMlnN
Power Property of Logarithms logaMp=plogaMlogaMp=plogaM lnMp=plnMlnMp=plnM

Now that we have the properties we can use them to “expand” a logarithmic expression. This means to write the logarithm as a sum or difference and without any powers.

We generally apply the Product and Quotient Properties before we apply the Power Property.

Example 10.33

Use the Properties of Logarithms to expand the logarithm log4(2x3y2)log4(2x3y2). Simplify, if possible.

Try It 10.65

Use the Properties of Logarithms to expand the logarithm log2(5x4y2)log2(5x4y2). Simplify, if possible.

Try It 10.66

Use the Properties of Logarithms to expand the logarithm log3(7x5y3)log3(7x5y3). Simplify, if possible.

When we have a radical in the logarithmic expression, it is helpful to first write its radicand as a rational exponent.

Example 10.34

Use the Properties of Logarithms to expand the logarithm log2x33y2z4log2x33y2z4. Simplify, if possible.

Try It 10.67

Use the Properties of Logarithms to expand the logarithm log4x42y3z25log4x42y3z25. Simplify, if possible.

Try It 10.68

Use the Properties of Logarithms to expand the logarithm log3x25yz3log3x25yz3. Simplify, if possible.

The opposite of expanding a logarithm is to condense a sum or difference of logarithms that have the same base into a single logarithm. We again use the properties of logarithms to help us, but in reverse.

To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.

Example 10.35

Use the Properties of Logarithms to condense the logarithm log43+log4xlog4ylog43+log4xlog4y. Simplify, if possible.

Try It 10.69

Use the Properties of Logarithms to condense the logarithm log25+log2xlog2ylog25+log2xlog2y. Simplify, if possible.

Try It 10.70

Use the Properties of Logarithms to condense the logarithm log36log3xlog3ylog36log3xlog3y. Simplify, if possible.

Example 10.36

Use the Properties of Logarithms to condense the logarithm 2log3x+4log3(x+1)2log3x+4log3(x+1). Simplify, if possible.

Try It 10.71

Use the Properties of Logarithms to condense the logarithm 3log2x+2log2(x1)3log2x+2log2(x1). Simplify, if possible.

Try It 10.72

Use the Properties of Logarithms to condense the logarithm 2logx+2log(x+1)2logx+2log(x+1). Simplify, if possible.

Use the Change-of-Base Formula

To evaluate a logarithm with any other base, we can use the Change-of-Base Formula. We will show how this is derived.

Suppose we want to evaluate logaMlogaM. logaMlogaM
Let y=logaMy=logaM. y=logaMy=logaM
Rewrite the expression in exponential form. ay=May=M
Take the logblogb of each side. logbay=logbMlogbay=logbM
Use the Power Property. ylogba=logbMylogba=logbM
Solve for yy. y=logbMlogbay=logbMlogba
Substitute y=logaMy=logaM. logaM=logbMlogbalogaM=logbMlogba

The Change-of-Base Formula introduces a new base b.b. This can be any base b we want where b>0,b1.b>0,b1. Because our calculators have keys for logarithms base 10 and base e, we will rewrite the Change-of-Base Formula with the new base as 10 or e.

Change-of-Base Formula

For any logarithmic bases a,ba,b and M>0,M>0,

logaM=logbMlogbalogaM=logMlogalogaM=lnMlna new basebnew base 10new baseelogaM=logbMlogbalogaM=logMlogalogaM=lnMlna new basebnew base 10new basee

When we use a calculator to find the logarithm value, we usually round to three decimal places. This gives us an approximate value and so we use the approximately equal symbol (≈)(≈).

Example 10.37

Rounding to three decimal places, approximate log435.log435.

Try It 10.73

Rounding to three decimal places, approximate log342.log342.

Try It 10.74

Rounding to three decimal places, approximate log546.log546.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with using the properties of logarithms.

Section 10.4 Exercises

Practice Makes Perfect

Use the Properties of Logarithms

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

218.

log41log41 log88log88

219.

log121log121 lnelne

220.

3log363log36 log227log227

221.

5log5105log510 log4410log4410

222.

8log878log87 log66−2log66−2

223.

6log6156log615 log88−4log88−4

224.

10log510log5 log10−2log10−2

225.

10log310log3 log10−1log10−1

226.

eln4eln4 lne2lne2

227.

eln3eln3 lne7lne7

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

228.

log46xlog46x

229.

log58ylog58y

230.

log232xylog232xy

231.

log381xylog381xy

232.

log100xlog100x

233.

log1000ylog1000y

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

234.

log338log338

235.

log656log656

236.

log416ylog416y

237.

log5125xlog5125x

238.

logx10logx10

239.

log10,000ylog10,000y

240.

lne33lne33

241.

lne416lne416

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

242.

log3x2log3x2

243.

log2x5log2x5

244.

logx−2logx−2

245.

logx−3logx−3

246.

log4xlog4x

247.

log5x3log5x3

248.

lnx3lnx3

249.

lnx43lnx43

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

250.

log5(4x6y4)log5(4x6y4)

251.

log2(3x5y3)log2(3x5y3)

252.

log3(2x2)log3(2x2)

253.

log5(214y3)log5(214y3)

254.

log3xy2z2log3xy2z2

255.

log54ab3c4d2log54ab3c4d2

256.

log4x16y4log4x16y4

257.

log3x2327y4log3x2327y4

258.

log22x+y2z2log22x+y2z2

259.

log33x+2y25z2log33x+2y25z2

260.

log25x32y2z44log25x32y2z44

261.

log53x24y3z3log53x24y3z3

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

262.

log64+log69log64+log69

263.

log4+log25log4+log25

264.

log280log25log280log25

265.

log336log34log336log34

266.

log34+log3(x+1)log34+log3(x+1)

267.

log25log2(x1)log25log2(x1)

268.

log73+log7xlog7ylog73+log7xlog7y

269.

log52log5xlog5ylog52log5xlog5y

270.

4log2x+6log2y4log2x+6log2y

271.

6log3x+9log3y6log3x+9log3y

272.

log3(x21)2log3(x1)log3(x21)2log3(x1)

273.

log(x2+2x+1)2log(x+1)log(x2+2x+1)2log(x+1)

274.

4logx2logy3logz4logx2logy3logz

275.

3lnx+4lny2lnz3lnx+4lny2lnz

276.

13logx3log(x+1)13logx3log(x+1)

277.

2log(2x+3)+12log(x+1)2log(2x+3)+12log(x+1)

Use the Change-of-Base Formula

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

278.

log342log342

279.

log546log546

280.

log1287log1287

281.

log1593log1593

282.

log217log217

283.

log321log321

Writing Exercises

284.

Write the Product Property in your own words. Does it apply to each of the following? loga5x,loga5x,loga(5+x).loga(5+x). Why or why not?

285.

Write the Power Property in your own words. Does it apply to each of the following? logaxp,logaxp,(logax)r.(logax)r. Why or why not?

286.

Use an example to show that
log(a+b)loga+logb.log(a+b)loga+logb.

287.

Explain how to find the value of log715log715 using your calculator.

Self Check

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

This table has three rows and four 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 use the properties of logarithms and use the change of base formula. The rest of the cells are blank.

On a scale of 110,110, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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