LOGARITHMIC FUNCTIONS

MATH 14 – PLANE TRIGONOMETRY

Logarithmic Functions

The logarithmic function f with base b, defined by

is the inverse of the exponential function with base b.

Logarithmic Functions

Properties of

The y-axis is the vertical asymptote of the graph.

The x-intercept of the graph is 1.

The graph is increasing when b > 1, while the graph is decreasing if 0 < b < 1.

The function is one-to-one.

Solving Logarithms

Given the expression logbx, we are looking for the

exponent of b that will give x.

General Form

The vertical asymptote (V.A.) of the graph is .

khxcaxgy b log

| 0 :g gD x c x h R

x h

How to sketch the graph?

1. Find the vertical asymptote. How?

2. Determine two arbitrary points. What points?

Use and

3. Locate the points.

4. Use the asymptote-two-point technique!

1hxc bhxc

x h

Illustrations

Illustrations

The number

In mathematics, is an irrational number

approximately equal to 2.718281846.

e

e

The natural expo and log functions

The natural exponential function is defined by

The natural logarithmic function is defined by

.xf x e

ln .f x x

Remarks:

The common logarithm:

logf x x

10log logx x

ln logex x

Logarithmic Expressions

xbxy y

b log

939log3 yy

252log5 xxy y

Finding the Inverse

Finding the inverse of an exponential function.

Finding the Inverse

Finding the inverse of a logarithmic function.

Finding the Inverse

Continuation…

TRY THIS Find the domain, range, V.A. and sketch the graph of

the following.

1.

2.

3.

12log2 3 xxg