Intermediate Algebra (11th edition)

604 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

OBJECTIVES Logarithms are important in many applications in biology, engineering, economics,

and social science. In this section we find numerical approximations for logarithms.

Traditionally, base 10 logarithms were used most often because our number system is

base 10. Logarithms to base 10 are called common logarithms,and

is abbreviated as

where the base is understood to be 10.

log 10 x log x,

Common and Natural Logarithms

10.5

1 Evaluate common

logarithms using

a calculator.

2 Use common

logarithms in

applications.

3 Evaluate natural

logarithms using

a calculator.

4 Use natural

logarithms in

applications.

5 Use the change-of-

base rule.

FIGURE 12 NOW TRY

In Example 1(c), a

negative result. The common logarithm of a num-

ber between 0 and 1 is always negativebecause

the logarithm is the exponent on 10 that produces

the number. In this case, we have

If the exponent (the logarithm) were positive, the

result would be greater than 1 because

The graph in FIGURE 13illustrates these concepts.

100 =1.

10 - 1.2111L0.0615.

log 0.0615L-1.2111,

OBJECTIVE 2 Use common logarithms in applications.In chemistry, pH is a

measure of the acidity or alkalinity of a solution. Pure water, for example, has pH 7.

In general, acids have pH numbers less than 7, and alkaline solutions have pH values

greater than 7, as shown in FIGURE 14on the next page.

x

y

(^0510)
–1
1
y = log 10 x
FIGURE 13

OBJECTIVE 1 Evaluate common logarithms using a calculator. In the first

example, we give the results of evaluating some common logarithms using a calcula-

tor with a LOG key. Consult your calculator manual to see how to use this key.

Evaluating Common Logarithms

Using a calculator, evaluate each logarithm to four decimal places.

(a) (b)

(c)

FIGURE 12shows how a graphing calculator displays these common logarithms to

four decimal places.

log 0.0615L-1.2111

log 327.1L2.5147 log 437,000L5.6405

EXAMPLE 1

NOW TRY
EXERCISE 1
Using a calculator, evaluate
each logarithm to four deci-
mal places.

(a)log 115 (b)log 0.25

NOW TRY ANSWERS

1. (a)2.0607 (b) -0.6021