Logarithmic Functions

1. Logarithmic Functions Essential Questions • How do we write equivalent forms for exponential and logarithmic functions? • How do we write, evaluate, and graph logarithmic functions? Holt McDougal Algebra 2 Holt Algebra 2

2. How many times would you have to double $1 before you had $512? You could solve this problem if you could solve 2x = 512 by using an inverse operation that undoes raising a base to an exponent equation to model this situation. This operation is called finding the logarithm. Alogarithm is the exponent to which a specified base is raised to obtain a given value.

3. Reading Math Read logba=x, as “the log base b of a is x.” Notice that the log is the exponent. You can write an exponential equation as a logarithmic equation and vice versa.

4. 1 1 log6 = –1 6 6 1 1 2 log255 = 2 Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. The base of the exponent becomes the base of the logarithm. log3243 = 5 1. The exponent is the logarithm. 2. 3. log1010,000 = 4 4. An exponent (or log) can be negative. logac =b The log (and the exponent) can be a variable. 5.

5. Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. The base of the exponent becomes the base of the logarithm. log981 = 2 6. log327 = 3 7. The exponent of the logarithm. The log (and the exponent) can be a variable. logx1 = 0 8.

6. 1 1 3 3 8 = 2 1 4–2 = 16 Converting from Logarithmic to Exponential Form Write each logarithmic form in exponential equation. The base of the logarithm becomes the base of the power. 91 = 9 9. 29 = 512 10. The logarithm is the exponent. 11. A logarithm can be a negative number. 1 12. 16 Any nonzero base to the zero power is 1. 13. b0 = 1

7. 1 1 2 2 –3 = 8 Converting from Logarithmic to Exponential Form Write each logarithmic form in exponential equation. The base of the logarithm becomes the base of the power. 14. 101 = 10 122 = 144 15. The logarithm is the exponent. An logarithm can be negative. 16.

8. A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log105. You can use mental math to evaluate some logarithms.

9. Evaluating Logarithms by Using Mental Math Evaluate by using mental math. = x 17. log 0.01 10x= 0.01 The log is the exponent. 10–2= 0.01 Think: What power of 10 is 0.01? log 0.01 = –2

10. Evaluating Logarithms by Using Mental Math Evaluate by using mental math. = x 18. log5 125 5x= 125 The log is the exponent. 53= 125 Think: What power of 5 is 125? log5125 = 3

11. 1 1 5 5 5x= 5–1= 1 log5 = –1 1 5 5 Evaluating Logarithms by Using Mental Math Evaluate by using mental math. = x 19. log5 The log is the exponent. Think: What power of 5 is ?

12. Evaluating Logarithms by Using Mental Math Evaluate by using mental math. 20. log 0.00001 = x 10x= 0.00001 The log is the exponent. 10–5= 0.01 Think: What power of 10 is 0.01? log 0.00001 = –5

13. Evaluating Logarithms by Using Mental Math Evaluate by using mental math. 21. log25 0.04 = x 25x= 0.04 The log is the exponent. 25–1= 0.04 Think: What power of 25 is 0.04? log250.04 = –1

14. Lesson 8.3 Practice A