Exponential and Logarithmic Functions 5

5.3 Logarithms 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 b t = r is called the logarithm of r with base b and is denoted by log b r.

Logarithms According to Definition 5.2, the logarithm of 16 base 2 is the exponent t such that 2 t = 16; thus we can write log 2 16 = 4. Likewise, we can write log = 3 because 10 3 = In general, Definition 5.2 can be remembered in terms of the statement log b r = t is equivalent to b t = r

Logarithms Evaluate log Example 1

Logarithms Solution: Let log = x. Changing to exponential form yields 10 x = , which can be solved as follows: 10 x = x = x = -4 Thus we have log = -4. Example 1

Properties of Logarithms Property 5.3 For b > 0 and b 1, log b b = 1 and log b 1 = 0

Properties of Logarithms Property 5.4 For b > 0, b 1, and r > 0, b log b r = r

Properties of Logarithms Property 5.5 For positive numbers b, r, and s, where b 1, log b rs = log b r + log b s

Properties of Logarithms If log 2 5 = and log 2 3 = , evaluate log Example 5

Properties of Logarithms Solution: Because 15 = 5 · 3, we can apply Property 5.5 as follows: log 2 15 = log 2 (5 · 3) = log log 2 3 = = Example 5

Properties of Logarithms Property 5.6 For positive numbers b, r, and s, where b 1,

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 log b r p = p(log b r)