Holt McDougal Algebra 2 4-4 Properties of Logarithms Warm Up 2. (3 –2 )(3 5 ) 1. (2 6 )(2 8 ) 3....

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Warm Up

2. (3–2)(35)1. (26)(28)

3. 4.

5. (73)5

Simplify.

Write in exponential form.

6. logx x = 1 7. 0 = log

x1

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Use properties to simplify logarithmic expressions. Translate between logarithms in any base.

Objectives

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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: log j + log a + log m = log j a mHelpful Hint

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Express as a single logarithm. Simplify, if possible.

log5625 + log525

Check It Out! Example 1a

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Express as a single logarithm. Simplify, if possible.

Check It Out! Example 1b

log 27 + log13

13

19

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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.

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Express log749 – log77 as a single logarithm.

Simplify, if possible.

log749 – log77

Check It Out! Example 2

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Express as a product. Simplify, if possibly.

a. log104 b. log5252

Check It Out! Example 3

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Express as a product. Simplify, if possibly.

c. log2 ( )5 1

2

Check It Out! Example 3

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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

Holt McDougal Algebra 2

4-4 Properties of Logarithms

b. Simplify 2log2(8x)a. Simplify log100.9

Check It Out! Example 4

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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.

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Evaluate log927.

Method 1 Change to base 10.

log927 =

Check It Out! Example 5a

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Evaluate log927.

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

Check It Out! Example 5a Continued

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Evaluate log816.

Method 1 Change to base 10.

Check It Out! Example 5b

Holt McDougal Algebra 2

4-4 Properties of Logarithms

Evaluate log816.

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

Check It Out! Example 5b Continued

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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.

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

Helpful Hint

Holt McDougal Algebra 2

4-4 Properties of Logarithms

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

Check It Out! Example 6

Holt McDougal Algebra 2

4-4 Properties of LogarithmsCheck It Out! Example 6 Continued