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7-4

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7-4

Properties of Logarithms

Warm Up

Lesson Presentation

Lesson Quiz

Holt McDougal Algebra 2

Holt Algebra 2

Warm Up

Simplify.

1. (26)(28)

214

33

2. (3–2)(35)

44

38

3.

4.

715

5. (73)5

Write in exponential form.

6. logxx = 1

7. 0 = logx1

x1 = x

x0 = 1

Objectives

Use properties to simplify logarithmic expressions.

Translate between logarithms in any base.

The logarithmic function for pH that you saw in the previous lessons, pH =–log[H+], can also be expressed in exponential form, as 10–pH= [H+].

Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents

Remember that to multiply powers with the same base, you add exponents.

The property in the previous slide can be used in reverse to write a sum of logarithms (exponents) as a single logarithm, which can often be simplified.

Helpful Hint

Think: logj+ loga+ logm = logjam

Example 1: Adding Logarithms

Express log64 + log69 as a single logarithm. Simplify.

log64 + log69

To add the logarithms, multiply the numbers.

log6 (4 9)

log6 36

Simplify.

Think: 6? = 36.

2

Check It Out! Example 1a

Express as a single logarithm. Simplify, if possible.

log5625 + log525

To add the logarithms, multiply the numbers.

log5 (625 • 25)

log5 15,625

Simplify.

6

Think: 5? = 15625

1

1

1

1

3

3

3

1

9

log27 + log

9

log(27 • )

log 3

1

1

3

3

Think: ? = 3

Check It Out! Example 1b

Express as a single logarithm. Simplify, if possible.

To add the logarithms, multiply the numbers.

Simplify.

–1

Remember that to divide powers with the same base, you subtract exponents

Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

Caution

Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.

The property above can also be used in reverse.

Example 2: Subtracting Logarithms

Express log5100 – log54 as a single logarithm. Simplify, if possible.

log5100 – log54

To subtract the logarithms, divide the numbers.

log5(100 ÷4)

Simplify.

log525

2

Think: 5? = 25.

Check It Out! Example 2

Express log749 – log77 as a single logarithm. Simplify, if possible.

log749 – log77

To subtract the logarithms, divide the numbers

log7(49 ÷ 7)

log77

Simplify.

1

Think: 7? = 7.

Because you can multiply logarithms, you can also take powers of logarithms.

Because 8 = 4, log84 = .

20() =

40

2

2

2

3

3

3

3

Example 3: Simplifying Logarithms with Exponents

Express as a product. Simplify, if possible.

A. log2326

B. log8420

6log232

20log84

Because 25 = 32, log232 = 5.

6(5) = 30

Check It Out! Example 3

Express as a product. Simplify, if possibly.

a. log104

b. log5252

4log10

2log525

Because 101 = 10, log10 = 1.

Because 52 = 25, log525 = 2.

4(1) = 4

2(2) = 4

5log2 ( )

Because 2–1 = , log2 = –1.

1

1

2

2

1

1

2

2

Check It Out! Example 3

Express as a product. Simplify, if possibly.

c. log2 ( )5

5(–1) = –5

Exponential and logarithmic operations undo each other since they are inverse operations.

Example 4: Recognizing Inverses

Simplify each expression.

c. 5log510

a. log3311

b. log381

log3311

log33 3 3 3

5log510

log334

10

11

4

Check It Out! Example 4

a. Simplify log100.9

b. Simplify 2log2(8x)

log 100.9

2log2(8x)

8x

0.9

Most calculators calculate logarithms only in base 10 or base e (see Lesson 7-6). You can change a logarithm in one base to a logarithm in another base with the following formula.

log8

log328 =

log32

≈

0.903

1.51

Example 5: Changing the Base of a Logarithm

Evaluate log328.

Method 1 Change to base 10

Use a calculator.

Divide.

≈ 0.6

log28

log328 =

3

log232

=

5

Example 5 Continued

Evaluate log328.

Method 2 Change to base 2, because both 32 and 8 are powers of 2.

Use a calculator.

= 0.6

log27

log927 =

log9

≈

1.431

0.954

Check It Out! Example 5a

Evaluate log927.

Method 1 Change to base 10.

Use a calculator.

≈ 1.5

Divide.

log327

log927 =

3

log39

=

2

Check It Out! Example 5a Continued

Evaluate log927.

Method 2 Change to base 3, because both 27 and 9 are powers of 3.

Use a calculator.

= 1.5

log16

Log816 =

log8

≈

1.204

0.903

Check It Out! Example 5b

Evaluate log816.

Method 1 Change to base 10.

Use a calculator.

Divide.

≈ 1.3

log416

log816 =

2

log48

=

1.5

Check It Out! Example 5b Continued

Evaluate log816.

Method 2 Change to base 4, because both 16 and 8 are powers of 2.

Use a calculator.

= 1.3

Helpful Hint

The Richter scale is logarithmic, so an increase of 1 corresponds to a release of 10 times as much energy.

Logarithmic scales are useful for measuring quantities that have a very wide range of values, such as the intensity (loudness) of a sound or the energy released by an earthquake.

Example 6: Geology Application

The tsunami that devastated parts of Asia in December 2004 was spawned by an earthquake with magnitude 9.3 How many times as much energy did this earthquake release compared to the 6.9-magnitude earthquake that struck San Francisco in1989?

The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula.

Substitute 9.3 for M.

Multiply both sides by .

æ

ö

E

13.95 = log

ç

÷

è

11.8

10

ø

3

2

Example 6 Continued

Simplify.

Apply the Quotient Property of Logarithms.

Apply the Inverse Properties of Logarithms and Exponents.

Example 6 Continued

Given the definition of a logarithm, the logarithm is the exponent.

Use a calculator to evaluate.

The magnitude of the tsunami was 5.6 1025 ergs.

Multiply both sides by .

3

2

Example 6 Continued

Substitute 6.9 for M.

Simplify.

Apply the Quotient Property of Logarithms.

5.6 1025

The tsunami released = 4000 times as much energy as the earthquake in San Francisco.

1.4 1022

Example 6 Continued

Apply the Inverse Properties of Logarithms and Exponents.

Given the definition of a logarithm, the logarithm is the exponent.

Use a calculator to evaluate.

The magnitude of the San Francisco earthquake was 1.4 1022 ergs.

Multiply both sides by .

3

2

Check It Out! Example 6

How many times as much energy is released by an earthquake with magnitude of 9.2 by an earthquake with a magnitude of 8?

Substitute 9.2 for M.

Simplify.

Check It Out! Example 6 Continued

Apply the Quotient Property of Logarithms.

Apply the Inverse Properties of Logarithms and Exponents.

Given the definition of a logarithm, the logarithm is the exponent.

Use a calculator to evaluate.

The magnitude of the earthquake is 4.0 1025 ergs.

Multiply both sides by .

3

2

Check It Out! Example 6 Continued

Substitute 8.0 for M.

Simplify.

Check It Out! Example 6 Continued

Apply the Quotient Property of Logarithms.

Apply the Inverse Properties of Logarithms and Exponents.

Given the definition of a logarithm, the logarithm is the exponent.

Use a calculator to evaluate.

The earthquake with a magnitude 9.2 released was ≈ 63 times greater.

4.0 1025

6.3 1023

Check It Out! Example 6 Continued

The magnitude of the second earthquake was 6.3 1023 ergs.

7

6

Lesson Quiz: Part I

Express each as a single logarithm.

1. log69 + log624

log6216 = 3

log327 = 3

2. log3108 – log34

Simplify.

3. log2810,000

30,000

4. log44x –1

x – 1

5. 10log125

125

6. log64128

8. log 10

1

2

Lesson Quiz: Part II

Use a calculator to find each logarithm to the nearest thousandth.

2.727

7. log320

–3.322

9. How many times as much energy is released by a magnitude-8.5 earthquake as a magntitude-6.5 earthquake?

1000