Holt McDougal Algebra 2

Properties of LogarithmsProperties of Logarithms

Holt Algebra 2Holt McDougal Algebra 2

•How do we use properties to simplify logarithmic

expressions?

•How do we translate between logarithms in any base?

Holt McDougal Algebra 2

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

Properties of Logarithms

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

Holt McDougal Algebra 2

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 jam

Helpful Hint

Holt McDougal Algebra 2

Properties of LogarithmsAdding Logarithms

2

To add the logarithms, multiply the numbers.

1. log64 + log

69

log6 (4 9)

log6 36 Simplify.

Think: 6 x = 36.

Express as a single logarithm. Simplify, if possible.

= x

Holt McDougal Algebra 2

Properties of LogarithmsAdding Logarithms

6

To add the logarithms, multiply the numbers.

2. log5625 + log525

log5 (625 • 25)

log5 15,625 Simplify.

Think: 5 x = 15625.

Express as a single logarithm. Simplify, if possible.

= x

Holt McDougal Algebra 2

Properties of LogarithmsAdding Logarithms

–1

To add the logarithms, multiply the numbers.

Simplify.

Express as a single logarithm. Simplify, if possible.

= x

9

1log27log .3

3

1

3

1

9

127log

3

1

3log3

1

Think: x = 313

Holt McDougal Algebra 2

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

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

Properties of LogarithmsSubtracting Logarithms

2

To subtract the logarithms, divide the numbers.

Simplify.

Express as a single logarithm. Simplify, if possible.

= x

4log100log .4 55

4

100log5

25log5

Think: 5 x = 25.

Holt McDougal Algebra 2

Properties of LogarithmsSubtracting Logarithms

1

To subtract the logarithms, divide the numbers.

Simplify.

Express as a single logarithm. Simplify, if possible.

= x

7log49log .5 77

7

49log7

7log7

Think: 7 x = 7.

Holt McDougal Algebra 2

Properties of Logarithms

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

Holt McDougal Algebra 2

Properties of LogarithmsSimplifying Logarithms with Exponents

30

Powers expressed as multiplication.

Simplify.

Express as a product. Simplify, if possible.6

232log .6

32log6 2

56 Think: 2 x = 32.

Holt McDougal Algebra 2

Properties of LogarithmsSimplifying Logarithms with Exponents

10

Powers expressed as multiplication.

Simplify.

Express as a product. Simplify, if possible.20

16 4log .7

4log20 16

2

120 Think: 16

x = 4.

Holt McDougal Algebra 2

Properties of LogarithmsSimplifying Logarithms with Exponents

4

Powers expressed as multiplication.

Simplify.

Express as a product. Simplify, if possible.401log .8

10log4

14 Think: 10 x = 10.

Holt McDougal Algebra 2

Properties of LogarithmsSimplifying Logarithms with Exponents

4

Powers expressed as multiplication.

Simplify.

Express as a product. Simplify, if possible.2

5 25log .9

25log2 5

22 Think: 5 x = 25.

Holt McDougal Algebra 2

Properties of LogarithmsSimplifying Logarithms with Exponents

5

Powers expressed as multiplication.

Simplify.

Express as a product. Simplify, if possible.5

2 2

1log .10

2

1log5 2

15 Think: 2 x = ½ .

Holt McDougal Algebra 2

Properties of Logarithms

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

Holt McDougal Algebra 2

Properties of LogarithmsRecognizing Inverses

Simplify each expression.

12. log381 13. 5log510

11. log3311

log3311

11

log334

4

5log510

10

14. log100.9

Log10

100.9

0.9

15. 2log2(8x)

2log2(8x)

8x

Holt McDougal Algebra 2

Properties of Logarithms

Most calculators calculate logarithms only in base 10 or base e. You can change a logarithm in one base to a logarithm in another base with the following formula.

Holt McDougal Algebra 2

Properties of LogarithmsChanging the Base of a Logarithm

16. Evaluate log32

8.

Change to base 10

log32

8 = log8log32 ≈ 0.6

17. Evaluate log927.

log927 = log27

log9 ≈ 1.5

Change to base 10

18. Evaluate log816.

log816 = log16

log8 ≈ 1.3

Change to base 10

Holt McDougal Algebra 2

Properties of Logarithms

Lesson 9.1 Practice A