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7-3 Logarithmic Functions Objectives & Vocabulary Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions. logarithm logarithmic function Holt Algebra 2 common logarithm 7-3 Logarithmic Functions Notes #1-2 Rewrite in other form Exponential Equation Logarithmic Form Logarithmic Form a. 92= 81 log101000 = 3 b. 33 = 27 log12144 = 2 0 x = 1(x ≠ 0) c. log 8 = –3 3A. Change 64 = 1296 to logarithmic form B. Change log279 = 2 to exponential form. 3 Calculate (without a calculator). 4. log864 Holt Algebra 2 1 5. log3 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 The base of the logarithm becomes the base of the power. 1 log99 = 1 9 =9 log2512 = 9 29 = 512 log82 = log4 1 16 = –2 logb1 = 0 Holt Algebra 2 1 3 The logarithm is the exponent. 1 3 8 =2 4 –2 = 1 16 b0 = 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 log3243 = 5 1 2 25 = 5 log255 = 104 = 10,000 6–1 = ab = c Holt Algebra 2 1 6 1 2 The base of the exponent becomes the base of the logarithm. The exponent is the logarithm. log1010,000 = 4 log6 1 6 = –1 logac =b 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 log981 = 2 The base of the exponent becomes the base of the logarithm. b. 33 = 27 log327 = 3 The exponent of the logarithm. logx1 = 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 log101000 = 3 103 = 1000 log12144 = 2 log 1 8 = –3 2 Holt Algebra 2 122 = 144 1 2 The base of the logarithm becomes the base of the power. The logarithm is the exponent. –3 =8 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 = log105. You can use mental math to evaluate some logarithms. Holt Algebra 2 7-3 Logarithmic Functions Example 3A: 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 3B: Evaluating Logarithms by Using Mental Math Evaluate without a calculator. log5 125 5? = 125 log5125 = 3 Holt Algebra 2 The log is the exponent. 7-3 Logarithmic Functions Example 3C/3D: Evaluating Logarithms by Using Mental Math Evaluate without a calculator. 3c. log5 1 log5 5 1 5 = –1 3d. log250.04 log250.04 = –1 Holt Algebra 2 7-3 Logarithmic Functions Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2x, is a logarithmic function, such as y = log2x. You may notice that the domain and range of each function are switched. The domain of y = 2x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log2x is {x|x > 0}, and the range is all real numbers (R). Holt Algebra 2 7-3 Logarithmic Functions Example 4A: 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. f(x) = 1 2 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 4A 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) 3A. Change 64 = 1296 to logarithmic formlog61296 = 4 B. Change log279 = 2 to exponential form. 27 3 Calculate the following using mental math (without a calculator). 4. log864 2 5. log3 1 –3 27 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) = 3x Then graph its inverse. Describe the domain and range of the inverse function. D: {x > 0}; R: all real numbers Holt Algebra 2