Intermediate Algebra (11th edition)

OBJECTIVE 5 Use the change-of-base rule.We have used a calculator to ap-

proximate the values of common logarithms (base 10) and natural logarithms (base e).

However, some applications involve logarithms with other bases. For example, the

amount of crude oil (in millions of barrels) imported into the United States during the

years 1990–2008 can be approximated by the function

where represents 1990, represents 1991, and so on. (Source:U.S.

Energy Information Administration.) To use this function, we need to find a base 2

logarithm. The following rule is used to convert logarithms from one base to another.

x= 1 x= 2

ƒ 1 x 2 = 2014 +384.7 log 2 x,

608 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

Leonhard Euler (1707–1783)

The number eis named after

Euler.

Change-of-Base Rule

If and then the following is true.

loga x

logb x

logb a

a 7 0,aZ 1,b 7 0,bZ1, x 7 0,

NOTE Any positive number other than 1 can be used for base bin the change-of-

base rule. Usually the only practical bases are eand 10 because calculators generally

give logarithms only for these two bases.

To derive the change-of-base rule, let

Change to exponential form.

Since logarithmic functions are one-to-one, if all variables are positive and if

then

Take the logarithm on each side.

Power rule

Substitute for m.

loga x= Divide by logb a.

logb x

logb a

1 loga x 21 logb a 2 =logb x

m logb a=logb x

logb 1 am 2 =logb x

logb x=logb y.

x=y,

am= x

loga x= m

loga x= m.

Using the Change-of-Base Rule

Evaluate to four decimal places.

Use common logarithms and the change-of-base rule.

Use a calculator.

NOW TRY

log 5 12 =

log 12

log 5

L1.5440

log 5 12

EXAMPLE 7

NOTE Either common or natural logarithms can be used when applying the change-

of-base rule. Verify that the same value is found in Example 7if natural logarithms

are used.

NOW TRY
EXERCISE 7
Evaluate to four deci-
mal places.

log 8 60

NOW TRY ANSWER

7. 1.9690