Warm Up

2. (3–2)(35)214 33

38

1. (26)(28)

3. 4.

5. (73)5 715

44

Simplify.

Write in exponential form.

x0 = 16. logx x = 1 x1 = x 7. 0 = log

x1

Use properties to simplify logarithmic expressions.

Translate between logarithms in any base.

Objectives

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.

Think: logj + loga + logm = logjam

Helpful Hint

Express log64 + log

69 as a single logarithm.

Simplify.

Example 1: Adding Logarithms

2

To add the logarithms, multiply the numbers.

log64 + log

69

log6 (4 9)

log6 36 Simplify.

Think: 6? = 36.

Express as a single logarithm. Simplify, if possible.

6

To add the logarithms, multiply the numbers.

log5625 + log

525

log5 (625 • 25)

log5 15,625 Simplify.

Think: 5? = 15625

Check It Out! Example 1a

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.

The property above can also be used in reverse.

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

Caution

Express log5100 – log

54 as a single logarithm.

Simplify, if possible.

Example 2: Subtracting Logarithms

To subtract the logarithms, divide the numbers.

log5100 – log

54

log5(100 ÷

4)

2

log525 Simplify.

Think: 5? = 25.

Express log749 – log

77 as a single logarithm.

Simplify, if possible.

To subtract the logarithms, divide the numbers

log749 – log

77

log7(49 ÷ 7)

1

log77 Simplify.

Think: 7? = 7.

Check It Out! Example 2

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

Express as a product. Simplify, if possible.

Example 3: Simplifying Logarithms with Exponents

A. log2326 B. log

8420

6log232

6(5) = 30

20log84

20( ) = 40 3

2 3

Because 8 = 4, log

84 = . 2

3

2 3

Because 25 = 32, log

232 = 5.

Express as a product. Simplify, if possibly.

a. log104 b. log5252

4log10

4(1) = 4

2log525

2(2) = 4Because 52 = 25, log

525

= 2.

Because 101 = 10, log

10 = 1.

Check It Out! Example 3

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

Example 4: Recognizing Inverses

Simplify each expression.

b. log381 c. 5log510a. log

3311

log3311

11

log33 3 3 3

log334

4

5log510

10

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.

Example 5: Changing the Base of a Logarithm

Evaluate log32

8.

Method 1 Change to base 10

log32

8 = log8log32

0.903 1.51

≈

≈ 0.6

Use a calculator.

Divide.

Example 5 Continued

Evaluate log32

8.

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

= 0.6

log32

8 =

log28

log232

=3

5 Use a calculator.

Evaluate log927.

Method 1 Change to base 10.

log927 = log27

log9

1.4310.954

≈

≈ 1.5

Use a calculator.

Divide.

Check It Out! Example 5a

Evaluate log927.

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

= 1.5

log927 =

log327

log39

=3

2 Use a calculator.

Check It Out! Example 5a Continued

Homework!

Chapter 7 Section 4 Page 516 #1-18, 20-34