3.2 Logarithmic Functions and Their Graphs

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3.2 Logarithmic Functions and Their Graphs. Definition of Logarithmic Function. 2 3 = 8. Ex. 3 = log 2 8. Ex. log 2 8. = x. 2 x = 8. x = 3. Properties of Logarithms and Natural Logarithms. log a 1 = 0 log a a = 1 log a a x = x. ln 1 = 0 ln e = 1 ln e x = x. Ex.

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3.2

Logarithmic Functions

and Their Graphs

Definition of Logarithmic Function

23 = 8

Ex. 3 = log2 8

Ex. log28

= x

2x = 8

x = 3

Properties of Logarithms and Natural Logarithms

• loga 1 = 0
• loga a = 1
• loga ax = x
• ln 1 = 0
• ln e = 1
• ln ex = x

Ex.

Use the definition of logarithm to write in

logarithmic form.

Ex.

4x = 16

log4 16 = x

e2 = x

ln x = 2

Graph and find the domain of the following functions.

y = ln x

x y

-2

-1

0

1

2

3

4

.5

cannot take

the ln of a (-)

number or 0

0

ln 2 = .693

ln 3 = 1.098

ln 4 = 1.386

D: x > 0

ln .5 = -.693

Graph y = 2x

y = x

x y

-2

-1

0

1

2

2-2 =

2-1 =

1

2

4

The graph of y = log2 x

is the inverse of y = 2x.

The domain of y = b +/- loga (bx + c), a > 1 consists

of all x such that bx + c > 0, and the V.A. occurs when

bx + c = 0. The x-intercept occurs when bx + c = 1.

Ex. Find all of the above for y = log3 (x – 2). Sketch.

D: x – 2 > 0

D: x > 2

V.A. @ x = 2

x-int. x – 2 = 1

x = 3

(3,0)