LESSON PLANNING Vocabulary logarithmic function common logarithm Extra Resources Reteaching 5.2 Extra Practice 5.2 Assignment In-class practice: 1 5 Homework: 6 33 Math Applications Exercise 3 from pages 238 245 START UP Tell students that loudness is a measure of the energy of sound. Since the range of loudness and the corresponding measurements of energy are so large, scientists invented the decibel scale. An increase of 1 in the decibel scale corresponds to a ten-fold increase in the corresponding energy level. INSTRUCTION Ask students to perform the following problems on their graphing calculators: log 10, log 100, log 1,000, log 0.1. Ask students to determine the base of the log key shown on their calculator. (Answers will vary, depending on calculator.) R.E.A.C.T. Strategy Relating Help students to understand the inverse relationship between exponential functions and logarithmic functions by likening it to the inverse relationship between squaring and taking the square root. For example, f(x) = x is determined by asking the question: what number do you square to get x? Further, f(x) = log 2 x is translated as the power to which you raise 2 to get x? 206 Chapter 5 Exponential and Logarithmic Functions
10 db 10 10 db 10 db 10 db 10 INSTRUCTION Show students how to evaluate logarithms without formally converting them to exponential form. To compute log 9 243, guide students to ask themselves 9 to what power yields 243? Students often confuse problems such as log b 0 and log b 1. Help them to clarify that the value of b does not affect the answer to these problems and that the former has no answer and that the latter is always 0. Also, students often wonder if b can be a negative number. Answer their questions by reminding them that in an exponential function f(x) = bx, b > 0. To reinforce their understanding of the relationship between logarithmic functions and exponential functions, ask students to work in pairs to use their calculators to evaluate log 4 15 to at least four decimal places. Point out to students that using the log key will not be beneficial. 5.2 Logarithmic Functions 207
Answer to Critical Thinking No, it is not possible. If x = log 10 1,000, then 10 x = 1,000. However, a positive base written to a power cannot be negative, so there is no solution for x. INSTRUCTION Point out that the graphs of g(x) and h(x) are reflections over the line y = x. Also point out that each point on the graph of g(x) can be transposed to find a point on the graph of h(x) and vice versa. Reteaching 5.2 (CRM) 5 2 1 128 7 3 Enriching the Lesson Present students with an alternative method of graphing the function f(x) = log 5 x. Tell students to make a table of values for the function f(x) = 5 x instead. Tell students that they need to transpose the x and y values when they graph the points. Have students practice this method on the function y = log 0.75 x also. 208 Chapter 5 Exponential and Logarithmic Functions
Answer to Ongoing Assessment 1 25 1 5 WRAP UP To ensure mastery of objectives, students should be able to: Write the logarithmic function that is the inverse to a given exponential function. Evaluate a logarithmic expression. Sketch the graph of a logarithmic function. Assignment In-class practice: 1 5 Homework: 6 33 Math Applications Exercise 3 from pages 238 245 Think and Discuss Answers 1. They are inverse functions. 2. The domain is all positive real numbers. The range is all real numbers. 3. 0; 0 b = 1 4. x = b y 5. Answers will vary. Sample answer: the decibel scale of sound intensity 5.2 Logarithmic Functions 209
Practice and Problem Solving Additional Answers 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 27. Answers will vary. Sample answer: The error is in the line x 2 = 16. This line should be 16 x = 2; x = 0.25. 28. 210 Chapter 5 Exponential and Logarithmic Functions
Extra Practice 5.2 (CRM) 5 1 3 Answers to Math Applications Math Applications for this chapter are on pages 238 245. The notes and solutions shown below accompany the suggested applications to assign with this lesson. 3. a. m = 2.512 (13 7.7) = 2.512 5.3 132 b. m = 2.512 (5.5 + 2.9) 2.292 c. m = 2.512 ( 1.9 + 26.73) 8,560,271,491 d. m = 2.512 (0.7 + 12.6) 209,055 5.2 Logarithmic Functions 211