7 3 Logarithmic Functions Objectives Vocabulary Write equivalent

7 -3 Logarithmic Functions Objectives & Vocabulary Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions. logarithmic function Holt Algebra 2 common logarithm

7 -3 Logarithmic Functions Notes #1 -2 Rewrite in other form Exponential Equation Logarithmic Form a. 92= 81 log 101000 = 3 b. 33 = 27 log 12144 = 2 0 c. x = 1(x ≠ 0) log 8 = – 3 3 A. Change 64 = 1296 to logarithmic form B. Change log 279 = 2 to exponential form. 3 Calculate (without a calculator). 4. log 864 Holt Algebra 2 1 5. log 3 27 Exponential Equation

7 -3 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 Algebra 2

7 -3 Logarithmic Functions Example 1: Converting from Logarithmic to Exponential Form Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation 1 log 99 = 1 9 =9 log 2512 = 9 29 = 512 log 82 = log 4 1 16 = – 2 logb 1 = 0 Holt Algebra 2 1 3 The base of the logarithm becomes the base of the power. 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.

7 -3 Logarithmic Functions Example 2: Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 35 = 243 log 3243 = 5 1 2 25 = 5 log 255 = 104 = 10, 000 log 1010, 000 = 4 6– 1 = ab = c Holt Algebra 2 1 6 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.

7 -3 Logarithmic Functions Notes #1 Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form a. 9 = 81 log 981 = 2 The base of the exponent becomes the base of the logarithm. b. 33 = 27 log 327 = 3 The exponent of the logarithm. logx 1 = 0 The log (and the exponent) can be a variable. 2 0 c. x = 1(x ≠ 0) Holt Algebra 2

7 -3 Logarithmic Functions Notes #2 Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation log 101000 = 3 103 = 1000 log 12144 = 2 log 8 = – 3 1 2 Holt Algebra 2 122 = 144 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.

7 -3 Logarithmic Functions A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the last example. Holt Algebra 2

7 -3 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 Algebra 2

7 -3 Logarithmic Functions Example 3 A: Evaluating Logarithms by Using Mental Math Evaluate by without a calculator. log 0. 01 10? = 0. 01 The log is the exponent. 10– 2 = 0. 01 Think: What power of 10 is 0. 01? log 0. 01 = – 2 Holt Algebra 2

7 -3 Logarithmic Functions Example 3 B: Evaluating Logarithms by Using Mental Math Evaluate without a calculator. log 5 125 5? = 125 log 5125 = 3 Holt Algebra 2 The log is the exponent.

7 -3 Logarithmic Functions Example 3 C/3 D: Evaluating Logarithms by Using Mental Math Evaluate without a calculator. 3 c. log 5 1 log 5 5 1 5 = – 1 3 d. log 250. 04 = – 1 Holt Algebra 2

7 -3 Logarithmic Functions Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2 x, is a logarithmic function, such as y = log 2 x. You may notice that the domain and range of each function are switched. The domain of y = 2 x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log 2 x is {x|x > 0}, and the range is all real numbers (R). Holt Algebra 2

7 -3 Logarithmic Functions Example 4 A: Graphing Logarithmic Functions Use the x-values {– 2, – 1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function. 1 2 f(x) = x Graph f(x) = 12 x by using a table of values. x f(x) =( 1 2 Holt Algebra 2 ) x – 2 – 1 0 1 2 4 2 1 1 2 1 4

7 -3 Logarithmic Functions Example 4 A Continued To graph the inverse, f– 1(x) = log 1 x, by using a table of 2 values. x f – 1(x) =log 1 2 x 4 2 1 – 2 – 1 0 1 2 1 1 4 2 The domain of f– 1(x) is {x|x > 0}, and the range is R. Holt Algebra 2

7 -3 Logarithmic Functions Notes (continued) 3 A. Change 64 = 1296 to logarithmic form log 61296 = 4 B. Change log 279 = 2 3 to exponential form. 27 Calculate the following using mental math (without a calculator). 4. log 864 2 1 5. log 3 27 – 3 Holt Algebra 2 2 3 =9

7 -3 Logarithmic Functions Notes (graphing) 6. Use the x-values {– 1, 0, 1, 2} to graph f(x) = 3 x Then graph its inverse. Describe the domain and range of the inverse function. D: {x > 0}; R: all real numbers Holt Algebra 2