Download Logarithms

5
Exponential and Logarithmic Functions
Exponential and Logarithmic Functions
5.3
Logarithms
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
Example 1
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
1
1
0.0001 
 4  104
10,000 10
x = -4
Thus we have log10 0.0001 = -4.
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
Example 5
If log2 5 = 2.3219 and log2 3 = 1.5850,
evaluate log215.
Properties of Logarithms
Example 5
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
Properties of Logarithms
Property 5.6
For positive numbers b, r, and s, where b  1,
r 
logb    logb r  logb 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)