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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)