Exponential and Logarithmic Functions

5

5.3Logarithms

Exponential and Logarithmic Functions

Objectives• Switch between exponential and logarithmic form

of equations.• Evaluate logarithmic expressions.• Solve logarithmic equations.• Apply the properties of logarithms to simplify

expressions.

Logarithms

Definition 5.2

If r is any positive real number, then the unique exponent t such that bt = r is called the logarithm of r with base b and is denoted by logb r.

Logarithms

According to Definition 5.2, the logarithm of 16 base 2 is the exponent t such that 2t = 16; thus we can write log2 16 = 4. Likewise, we can write log10 1000 = 3 because 103 = 1000. In general, Definition 5.2 can be remembered in terms of the statement

logb r = t is equivalent to bt = r

Logarithms

Evaluate log10 0.0001.

Example 1

Logarithms

Solution:Let log10 0.0001 = x. Changing to exponential form yields 10x = 0.0001, which can be solved as follows:

10x = 0.0001

10x = 10-4

x = -4

Thus we have log10 0.0001 = -4.

Example 1

44

1 10.0001 10

10,000 10

Properties of Logarithms

Property 5.3

For b > 0 and b 1,logb b = 1 and logb 1 = 0

Properties of Logarithms

Property 5.4

For b > 0, b 1, and r > 0,blogb r = r

Properties of Logarithms

Property 5.5

For positive numbers b, r, and s, where b 1,logb rs = logb r + logb s

Properties of Logarithms

If log2 5 = 2.3219 and log2 3 = 1.5850, evaluate log215.

Example 5

Properties of Logarithms

Solution:

Because 15 = 5 · 3, we can apply Property 5.5 as follows:

log2 15 = log2(5 · 3)

= log2 5 + log2 3

= 2.3219 + 1.5850 = 3.9069

Example 5

Properties of Logarithms

Property 5.6

For positive numbers b, r, and s, where b 1,

log log logb b b

rr s

s

Properties of Logarithms

Property 5.7

If r is a positive real number, b is a positive real number other than 1, and p is any real number, then

logb rp = p(logb r)