Section 3.2

Logarithmic Functions

The Logarithmic Function

xxf alog)(

0a

1a

The Big Idea

•The logarithmic function is the inverse of the exponential function.

Important Equivalency

• Is equivalent to

xy alog

xa y

Convert From Logarithmic to Exponential

M

b

2

7

log2

49log2

343log3

Convert from Exponential to Logarithmic

502

401

464

5125

1

5

3

3

c

y

How to Find Logs

1. Use what you know. If you know that 2 to the 3rd power is 8, then you also know that the log, base 2, of 8 is 3.

2. Use properties of exponents and radicals. Taking the square root of something is the same as raising that same thing to the ½ power. And fractions can often be re-written using negative exponents.

Evaluate the Following Log Expressions

4log

8log

16

1log

49log

64

8

2

7

How to Find Logs, continued

3. Use these helpful log properties:

xb

xb

b

x

xb

b

b

b

log

log

1log

01log

Evaluate the Following Log Expressions

7log

3

7

44

1log

7log

The domain of a logarithmic function

• It is not possible to take the log of a negative number.

• To find the domain of a logarithmic function, set the “argument” > 0.

Find the domain of each logarithmic function

)13log()(

)7(log)( 4

xxf

xxg

Natural and Common Logs

• Logs to the base of 10 are called common logs (log on your calculator)

• Logs to the base of e are called natural logs (ln on your calculator)

Evaluate the Following Log Expressions

129ln

5

28log

ln

10

10

1log

e

e

Graphs of Logarithmic Functions

• You may omit questions dealing with graphs on both the homework (15 – 18) and the quiz (6 and 7). I will give you credit for those questions.