5 Exponential and Logarithmic Functions Exponential and Logarithmic

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 2 t = 16; thus we can write log 2 16 = 4. Likewise, we can write log 10 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 log 10 0. 0001. Example 1

Logarithms Example 1 Solution: Let log 10 0. 0001 = x. Changing to exponential form yields 10 x = 0. 0001, which can be solved as follows: 10 x = 0. 0001 10 x = 10 -4 x = -4 Thus we have log 10 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 logb rs = logb r + logb s 1,

Properties of Logarithms Example 5 If log 2 5 = 2. 3219 and log 2 3 = 1. 5850, evaluate log 215.

Properties of Logarithms Example 5 Solution: Because 15 = 5 · 3, we can apply Property 5. 5 as follows: log 2 15 = log 2(5 · 3) = log 2 5 + log 2 3 = 2. 3219 + 1. 5850 = 3. 9069

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 logb rp = p(logb r)