Properties of Logarithmic Functions

Objective: Simplify and evaluate expressions involving logarithms.

7z 12t 3c

7z 12t 3c

3

3

nm

66ba 48sr

Product and Quotient Properties of Logarithms

• For

• Product Property

:1,0,0,0 bbnm

nmmn bbb loglog)(log

Product and Quotient Properties of Logarithms

• For

• Product Property

• Quotient Property

:1,0,0,0 bbnm

nmmn bbb loglog)(log

nm bbnm

b logloglog

Product and Quotient Properties of Logarithms

• For

• Product Property

• Quotient Property

• You can use these properties to evaluate logarithmic expressions.

:1,0,0,0 bbnm

nmmn bbb loglog)(log

nm bbnm

b logloglog

Example 1

Example 1

Example 1

Try This

• Given that , approximate each expression.

585.13log2

18log2 43

2log

Try This

• Given that , approximate each expression.

585.13log2

18log2 43

2log

170.4585.1585.11

3log3log2log

332log

222

2

Try This

• Given that , approximate each expression.

585.13log2

18log2 43

2log

170.4585.1585.11

3log3log2log

332log

222

2

415.2585.1

4log3log 22

Example 2

Example 2

Example 2

Try This

• Write each expression as a single logarithm. Then, simplify if possible.

6log18log 44 yyx bb log3log4log 3

Try This

• Write each expression as a single logarithm. Then, simplify if possible.

6log18log 44 yyx bb log3log4log 3

3loglog 4618

4

Try This

• Write each expression as a single logarithm. Then, simplify if possible.

6log18log 44 yyx bbb log3log4log

3loglog 4618

4 34

34 loglog x

byyx

b

The Power Property of Logarithms

• For , and any real number p:1,0,0,0 bbnm

mpm bp

b loglog

Example 3

Example 3

Try This

• Evaluate .1003 27log

Try This

• Evaluate .1003 27log

300310027log10027log 3100

3

Exponential-Logarithmic Inverse Properties

• For b > 0 and : 1b

0for log xxb xb

0for log xxb xb

Example 4

Example 4

Example 4

Try This

• Evaluate each expression.

81log7 311log7 8log5

8338log

Try This

• Evaluate each expression.

81log7 311log7 8log5

8338log

7411

Try This

• Evaluate each expression.

81log7 311log7 8log5

8338log

7411 1385

One-to-One Property of Logarithms

• If , then x = y.yx bb loglog

Example 5

Example 5

Example 5

Homework

• Page 382• 13-61 odd