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Logarithmic Functions

Learn how to write equivalent forms, evaluate, and graph logarithmic functions. Understand the conversion from exponential to logarithmic form and vice versa. Practice using mental math to evaluate logarithms effectively.

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Logarithmic Functions

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  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

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