1 / 13

Exponential and Logarithmic Functions

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.

Télécharger la présentation

Exponential and Logarithmic Functions

E N D

Presentation Transcript

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)

More Related