Intermediate Algebra (11th edition)

This key statement should be memorized.The abbreviation logis used for the word

logarithm.Read as “the logarithm of xwith base a” or “the base aloga-

rithm of x.” To remember the location of the base and the exponent in each form,

refer to the following diagrams.

Exponent Exponent

Logarithmic form: Exponential form:

Base Base

In work with logarithmic form and exponential form, remember the following.

y=loga x x=a y

loga x

588 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

OBJECTIVES The graph of is the curve shown in blue in FIGURE 9. Because defines

a one-to-one function, it has an inverse. Interchanging xand ygives

the inverse of Roles of xand yare interchanged.

As we saw in Section 10.1,the graph of the inverse is found by reflecting the graph of

y= 2 xabout the line y=x.The graph of x= 2 yis shown as a red curve inFIGURE 9.

x= 2 y, y= 2 x.

y= 2 x y= 2 x

Logarithmic Functions

10.3

1 Define a logarithm.

2 Convert between

exponential and

logarithmic forms.

3 Solve logarithmic

equations of the

form for

a, b, or k.

4 Define and graph

logarithmic

functions.

5 Use logarithmic

functions in

applications

involving growth

or decay.

loga b=k

x

y

2

4

6

8

8

y = 2x

or y = log 2 x

x = 2y

2 4 6

y = x

FIGURE 9

OBJECTIVE 1 Define a logarithm. We cannot solve the equation for the

dependent variable ywith the methods presented up to now. The following definition

is used to solve x= 2 yfor y.

x= 2 y

Logarithm

For all positive numbers a, with and all positive numbers x,

yloga x means the same as xay.

aZ1,

Meaning of

A logarithm is an exponent. The expression represents the exponent to

which the base a must be raised to obtain x.

loga x

loga x

OBJECTIVE 2 Convert between exponential and logarithmic forms. We

can use the definition of logarithm to convert between exponential and logarithmic

forms.