Logarithmic Functions How do we write equivalent forms

Logarithmic. Functions • How do we write equivalent forms for exponential and logarithmic functions? • How do we write, evaluate, and graph logarithmic functions? Holt. Mc. Dougal Algebra 2 Holt

Logarithmic Functions How many times would you have to double $1 before you had $512? You could solve this problem if you could solve 2 x = 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. A logarithm is the exponent to which a specified base is raised to obtain a given value. Holt Mc. Dougal Algebra 2

Logarithmic Functions You can write an exponential equation as a logarithmic equation and vice versa. Reading Math Read logb a= x, as “the log base b of a is x. ” Notice that the log is the exponent. Holt Mc. Dougal Algebra 2

Logarithmic Functions Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. 1. Exponential Equation Logarithmic Form 35 = 243 log 3243 = 5 1 2 2. 25 = 5 log 255 = 3. 104 = 10, 000 log 1010, 000 = 4 4. 6 – 1 5. ab = c = 1 6 Holt Mc. Dougal Algebra 2 log 6 1 6 = – 1 logac =b The base of the exponent becomes the base of the logarithm. The exponent is the logarithm. An exponent (or log) can be negative. The log (and the exponent) can be a variable.

Logarithmic Functions Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 6. 9 = 81 log 981 = 2 The base of the exponent becomes the base of the logarithm. 7. 33 = 27 log 327 = 3 The exponent of the logarithm. logx 1 = 0 The log (and the exponent) can be a variable. 2 0 8. x = 1(x ≠ 0) Holt Mc. Dougal Algebra 2

Logarithmic Functions Converting from Logarithmic to Exponential Form Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation 9. log 99 = 1 9 =9 10. log 2512 = 9 29 = 512 11. log 82 = 12. 13. log 4 1 16 The base of the logarithm becomes the base of the power. 1 1 3 = – 2 logb 1 = 0 Holt Mc. Dougal Algebra 2 The logarithm is the exponent. 1 3 8 =2 4– 2 = 1 16 b 0 = 1 A logarithm can be a negative number. Any nonzero base to the zero power is 1.

Logarithmic Functions Converting from Logarithmic to Exponential Form Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation 14. log 1010 = 10 15. log 12144 = 2 122 = 144 16. log 8 = – 3 1 2 Holt Mc. Dougal Algebra 2 1 1 2 – 3 =8 The base of the logarithm becomes the base of the power. The logarithm is the exponent. An logarithm can be negative.

Logarithmic Functions 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 = log 105. You can use mental math to evaluate some logarithms. Holt Mc. Dougal Algebra 2

Logarithmic Functions Evaluating Logarithms by Using Mental Math Evaluate by using mental math. 17. log 0. 01 = x 10 x = 0. 01 The log is the exponent. 10– 2 = 0. 01 Think: What power of 10 is 0. 01? log 0. 01 = – 2 Holt Mc. Dougal Algebra 2

Logarithmic Functions Evaluating Logarithms by Using Mental Math Evaluate by using mental math. 18. log 125 = x 5 5 x = 125 The log is the exponent. 53 = 125 Think: What power of 5 is 125? log 5125 = 3 Holt Mc. Dougal Algebra 2

Logarithmic Functions Evaluating Logarithms by Using Mental Math Evaluate by using mental math. 19. log =x 5 5 x = 1 5 The log is the exponent. 1 5 5– 1 = 1 log 5 5 = – 1 Holt Mc. Dougal Algebra 2 Think: What power of 5 is 1 ? 5

Logarithmic Functions Evaluating Logarithms by Using Mental Math Evaluate by using mental math. 20. log 0. 00001 = x 10 x = 0. 00001 The log is the exponent. 10– 5 = 0. 01 Think: What power of 10 is 0. 01? log 0. 00001 = – 5 Holt Mc. Dougal Algebra 2

Logarithmic Functions Evaluating Logarithms by Using Mental Math Evaluate by using mental math. 21. log 0. 04 = x 25 25 x = 0. 04 The log is the exponent. 25– 1 = 0. 04 Think: What power of 25 is 0. 04? log 250. 04 = – 1 Holt Mc. Dougal Algebra 2

Logarithmic Functions Lesson 8. 3 Practice A Holt Mc. Dougal Algebra 2