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



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



Express as a single logarithm. Simplify, if possible.

log5625 + log525

Check It Out! Example 1a



Express as a single logarithm. Simplify, if possible.

Check It Out! Example 1b

log 27 + log13

13

19



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 log749 – log77 as a single logarithm.

Simplify, if possible.

log749 – log77

Check It Out! Example 2



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



Express as a product. Simplify, if possibly.

a. log104 b. log5252




Express as a product. Simplify, if possibly.

c. log2 ( )5 1

2




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



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




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.



Evaluate log927.

Method 1 Change to base 10.

log927 =

Check It Out! Example 5a



Evaluate log927.

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

Check It Out! Example 5a Continued



Evaluate log816.

Method 1 Change to base 10.

Check It Out! Example 5b



Evaluate log816.

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

Check It Out! Example 5b Continued



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



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



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

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

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