3.2Logarithmic Functions

and Their Graphs

Definition of Logarithmic Function

xy alog= xa y =

Ex. 3 = log2 8 23 = 8

Ex. log28 = x 2x = 8

x = 3

Properties of Logarithms and Natural Logarithms

1. loga 1 = 0

2. loga a = 1

3. loga ax = x

1. ln 1 = 0

2. ln e = 1

3. ln ex = x

Ex. =e

1ln =−1ln e 1−

=2ln e 2

Use the definition of logarithm to write inlogarithmic 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

-2-101234.5

cannot takethe ln of a (-) number or 00ln 2 = .693ln 3 = 1.098ln 4 = 1.386ln .5 = -.693

D: x > 0

Graph y = 2x

x y

-2-1012

2-2 = 4

1

2-1 =2

1

124

The graph of y = log2 x is the inverse of y = 2x.

y = x

The domain of y = b +/- loga (bx + c), a > 1 consistsof all x such that bx + c > 0, and the V.A. occurs whenbx + 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)