3.3 Logarithms and Logarithmic Functions For use with Eploration 3.3 Essential Question What are some of the characteristics of the graph of a logarithmic function? Every eponential function of the form f ( ) = b, where b is a positive real number other than, has an inverse function that you can denote by g( ) = log b. This inverse function is called a logarithmic function with base b. EXPLORATION: Rewriting Eponential Equations Work with a partner. Find the value of in each eponential equation. Eplain your reasoning. Then use the value of to rewrite the eponential equation in its equivalent logarithmic form, = log y. b a. 2 = 8 b. 3 = 9 c. 4 = 2 d. 5 = e. 5 = f. 8 = 4 5 2 EXPLORATION: Graphing Eponential and Logarithmic Functions Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Complete each table for the given eponential function. Use the results to complete the table for the given logarithmic function. Eplain your reasoning. Then sketch the graphs of f and g in the same coordinate plane. a. 2 0 2 f( ) = 2 g( ) = log 2 2 0 2 05
3.3 Logarithms and Logarithmic Functions (continued) 2 EXPLORATION: Graphing Eponential and Logarithmic Functions (continued) b. 2 0 2 f( ) = 0 g( ) = log 0 2 0 2 3 EXPLORATION: Characteristics of Graphs of Logarithmic Functions Work with a partner. Use the graphs you sketched in Eploration 2 to determine the domain, range, -intercept, and asymptote of the graph of g( ) = log, where b is a positive real number other than. Eplain your reasoning. b Communicate Your Answer 4. What are some of the characteristics of the graph of a logarithmic function? 5. How can you use the graph of an eponential function to obtain the graph of a logarithmic function? 06
3.3 Notetaking with Vocabulary For use after Lesson 3.3 In your own words, write the meaning of each vocabulary term. logarithm of y with base b common logarithm natural logarithm Core Concepts Definition of Logarithm with Base b Let b and y be positive real numbers with b. The logarithm of y with base b is denoted by log b y and is defined as log b y = if and only if b = y. The epression log b y is read as log base b of y. Notes: 07
Name Date 3.3 Notetaking with Vocabulary (continued) Parent Graphs for Logarithmic Functions f = is shown below for b > and for 0 < b <. The graph of ( ) log b Because f ( ) = log b and g( ) b f ( ) = log b is the reflection of the graph of g( ) b Graph of ( ) = logb f for > = are inverse functions, the graph of b Graph of ( ) log = in the line y =. f for 0 < b < = b y y g() = b (0, ) (, 0) g() = b (0, ) (, 0) f() = log b f() = log b Note that the y-ais is a vertical asymptote of the graph of f ( ) log b. f( ) = log is > 0, and the range is all real numbers. b = The domain of Notes: Practice A Etra Practice In Eercises 4, rewrite the equation in eponential form.. log0000 = 3 2. log5 2 25 = 3. log0 = 0 4. log464 = 3 08
3.3 Notetaking with Vocabulary (continued) In Eercises 5 8, rewrite the equation in logarithmic form. 5. 2 2 = 44 6. 3 20 = 7. 26 = 6 8. 20 0 4 = In Eercises 9 2, evaluate the logarithm. 9. log4 64 0. log8. log2 2. log25 32 5 In Eercises 3 and 4, simplify the epression. 3. log 6 3 3 4. ln e 3 In Eercises 5 and 6, find the inverse of the function. y = + 6. y ( ) 5. 5 0 = ln 2 8 In Eercises 7 and 8, graph the function. Determine the asymptote of the function. 7. y log ( ) y = log 4 = 2 + 8. 2 09
Practice 5.2 BPractice B In Eercises 3, rewrite the equation in eponential form.. log9 = 0 2. log6 26 = 3 3. log 2 = 2 4 In Eercises 4 6, rewrite the equation in logarithmic form. 4. 2 3 = 5. 69 32 4 = 8 6. 2 8 = 9 In Eercises 7 2, evaluate the logarithm. 7. log8 64 8. log2 32 9. log0 0. log 3. log2 0.25 2. log0 0.0 8 In Eercises 3 5, evaluate the logarithm using a calculator. Round your answer to three decimal places. 3. ( ) log 4. 2 ln(.4 ) 5. ( ) 5 ln 0.4 2 6. The decibel level D of sound is given by the equation 0 log I D =, 2 0 where I is the intensity of the sound. The pain threshold for sound is 25 decibels. Does a sound with an intensity of 0 eceed the pain threshold? Eplain. In Eercises 7 9, simply the epression. ln 7 log 8 7. e 8. In Eercises 20 25, find the inverse of the function. 3 0 9. log( 0 ) 20. y = 0.75 2. y = log 22. y = log 34 2 23. y ln( 2) = + 24. y 3 = e 25. y = 6 + 2 26. The length (in inches) of an alligator and its weight w (in pounds) are related by the function = 27. ln w 32.8. a. Estimate the length (in inches) of an alligator that weighs 250 pounds. What is its length in feet? b. Find the inverse of the given function. Use the inverse function to find the weight of a 4-foot alligator. (Hint: Convert to inches first.) 0