2.2 Logarithmic Functions MATH14

• Logarithmic Function with base b • Graph of Logarithmic Function • Natural Logarithmic Function • Properties of Logarithmic Functions • Exponential and Logarithmic Equations

Logarithmic Function with base b

Definition: The logarithmic function with base b is the inverse of the exponential function with base b.

y logb x

Note: Dom f

if and only if

x b

y

Rng f

Logarithmic Function with base b

Examples:

3 9 1/2 1 1 4 16

2

log 3 9 2 1 1 log1/16 4 2

2 8

3

1 5 25

2

Logarithmic Function with base b

More examples: Find the values of the ff:

Solution:

log 7 49

log 7 49 y 7 49

y y

Therefore, log 7 49 2

7 7

2

Logarithmic Function with base b

More examples: Find the values of the ff:

Solution: log 5 y 5

y y

log 5 5

5 5 5 5

1/2

Therefore,log 5

5 1/ 2

Logarithmic Function with base b

More examples: Find the values of the ff:

1 log 6 6 Solution: log 1/ 6 y 6

y y

6 1/ 6 6 6

1

Therefore,log 6 1/ 6 1

Logarithmic Function with base b

More examples: Find the values of the ff:

Solution:

log3 81 log 3 81 y

3 81

y y

3 3

Therefore,

4

log3 81 4

Logarithmic Function with base b

More examples: Find the values of the ff:

log10 0.001

Solution:

log10 0.001 y 10 y 0.001 10 10

y 3

Therefore,log10 0.001 3

Logarithmic Function with base b

Other examples:

Solve the given equation for either x or b.

log 6 x 2

6 x

2

Solution:

log 6 x 2 x 36

Logarithmic Function with base b

Other examples:

Solve the given equation for either x or b.

log 27 x 2 / 3

27 2/3 x x

Solution:

log 27 x 2 / 3

3

27

2

x9

Logarithmic Function with base b

Other examples:

Solve the given equation for either x or b.

logb 4 1/ 3

log b 4 1/ 3 b

1/3 1/3 3

Solution:

4 4

3

b

b 64

Logarithmic Function with base b

Other examples:

Solve the given equation for either x or b.

logb 81 2

log b 81 2 b 81

2

Solution:

b

2

1/2

81

1/2

b 1/ 9

Logarithmic Function with base b

Recall:

f f 1 x x f 1 f x x

Since the logarithmic function with base b is the inverse of exponential function with base b, then

log b b x

x

b

logb x

x

Logarithmic Function with base b

Example:

log 2 2 5 log10 10 3

3

5

Graph of Logarithmic function of base b

Since logarithmic function is the inverse of exponential function, then its graph is the reflection of the latter function at y=x.

b 1

y logb x

y b

x

0 b 1

y b

x

y logb x

Natural Logarithmic Function

Definition: The natural logarithmic function is the inverse of the natural exponential function.

y ln x

if and only if

xe

y

Dom f

Rng f

Natural Logarithmic Function

Note:

ln e 1

e

ln x

x

x

ln e x

Properties of Logarithmic Functions

Theorem: If b>0, b≠1, and u and v are positive numbers, then

logb uv logb u logb v

u log b log b u log b v v

Properties of Logarithmic Functions

Theorem: If b>0, b≠1, and u and v are positive numbers, then

logb u n logb u

n

Properties of Logarithmic Functions

Examples: Express each of the

following in terms of log of x, y and z, each of which represents a positive number.

log b x y z

2

3 4

x log b 2 yz

log b

5

xy 3 z

2

Properties of Logarithmic Functions

Examples: Write each of the

following expression as a single logarithm with a coefficient of 1.

logb x 2logb y 3logb z

1 logb 4 logb 3 logb x logb y 3

TOPICS for TODAY

• Exponential and Logarithmic Equations • Application • EXERCISE

RULES FOR SOLVING LOGARITHMIC AND EXPONENTIAL EQUATIONS • Use log of base 10...