Exponential and Logarithmic Functions

1. 5 Exponential and Logarithmic Functions

2. 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.

3. 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.

4. 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

5. Logarithms Example 1 Evaluate log10 0.0001.

6. 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 x = -4 Thus we have log10 0.0001 = -4.

7. Properties of Logarithms Property 5.3 For b > 0 and b 1, logb b = 1 and logb1 = 0

8. Properties of Logarithms Property 5.4 For b > 0, b 1, and r > 0, blogb r= r

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

10. Properties of Logarithms Example 5 If log2 5 = 2.3219 and log2 3 = 1.5850, evaluate log215.

11. 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

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

13. 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)