Apply Properties of Logarithms. Let b, m, and n be positive numbers such that b Þ 1. m 1 log b. mn 5 log b. m }n 5 log b. log b.

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1 TEKS 7.5 a.2, 2A.2.A, 2A.11.C Apply Properties of Logarithms Before You evaluated logarithms. Now You will rewrite logarithmic epressions. Why? So you can model the loudness of sounds, as in E. 63. Key Vocabulary base, p. 10 KEY CONCEPT Properties of Logarithms Let b, m, and n be positive numbers such that b Þ 1. Product Property log b mn 5 log b m 1 log b n For Your Notebook Quotient Property Power Property log b m }n 5 log b m 2 log b n log b m n 5 n log b m E XAMPLE 1 Use properties of logarithms AVOID ERRORS Note that in general log b m }n Þ log b m } logb n and log b mn Þ(log b m)(log b n). Use log 4 3 ø and log 4 7 ø to evaluate the logarithm. a. log 3 4 }7 5 log log 4 7 Quotient property ø Use the given values of log 4 3 and log Simplify. b. log log 4 (3 p 7) Write 21 as 3 p 7. 5 log log 4 7 Product property ø Use the given values of log 4 3 and log Simplify. c. log log Write 49 as log 4 7 Power property ø 2(1.404) Use the given value of log Simplify. GUIDED PRACTICE for Eample 1 Use log 6 5 ø and log 6 8 ø to evaluate the logarithm. 1. log 6 5 }8 2. log log log Apply Properties of Logarithms 507

2 REWRITING EXPRESSIONS You can use the properties of logarithms to epand and condense logarithmic epressions. REWRITE EXPRESSIONS When you are epanding or condensing an epression involving logarithms, you may assume any variables are positive. E XAMPLE 2 Epand log } y. Epand a logarithmic epression log } y 5 log log 6 y Quotient property 5 log log log 6 y Product property 5 log log 6 2 log 6 y Power property E XAMPLE 3 TAKS PRACTICE: Multiple Choice Which of the following is equivalent to log log 4 2 log 6? A log 6 B log 8 C log 32 D log 61 Solution log log 4 2 log 6 5 log 3 1 log log 6 Power property c The correct answer is C. A B C D 5 log (3 p 4 3 ) 2 log 6 Product property 5 log 3 p 43 } 6 Quotient property 5 log 32 Simplify. GUIDED PRACTICE for Eamples 2 and 3 5. Epand log Condense ln ln 3 2 ln 12. CHANGE-OF-BASE FORMULA Logarithms with any base other than 10 or e can be written in terms of common or natural logarithms using the change-of-base formula. This allows you to evaluate any logarithm using a calculator. KEY CONCEPT For Your Notebook Change-of-Base Formula If a, b, and c are positive numbers with b Þ 1 and c Þ 1, then: log c a 5 log b a } logb c In particular, log c a 5 log a } log c and log c a 5 ln a } ln c. 508 Chapter 7 Eponential and Logarithmic Functions

3 E XAMPLE 4 Use the change-of-base formula Evaluate log 3 8 using common logarithms and natural logarithms. Solution Using common logarithms: log 3 8 5} log 8 ø} ø log Using natural logarithms: log ln 8 } ln 3 ø } ø E XAMPLE 5 Use properties of logarithms in real life SOUND INTENSITY For a sound with intensity I (in watts per square meter), the loudness L(I) of the sound (in decibels) is given by the function L(I) 5 10 log I } I0 where I 0 is the intensity of a barely audible sound (about watts per square meter). An artist in a recording studio turns up the volume of a track so that the sound s intensity doubles. By how many decibels does the loudness increase? Solution Let I be the original intensity, so that 2I is the doubled intensity. Increase in loudness 5 L(2I) 2 L(I) Write an epression log 2I } I log I } I0 Substitute log 2I } 2 log } 1 I Distributive property I0 I log 2 1 log I } I0 2 log I } I0 2 Product property 5 10 log 2 Simplify. ø 3.01 c The loudness increases by about 3 decibels. Use a calculator. GUIDED PRACTICE for Eamples 4 and 5 Use the change-of-base formula to evaluate the logarithm. 7. log log log log WHAT IF? In Eample 5, suppose the artist turns up the volume so that the sound s intensity triples. By how many decibels does the loudness increase? 7.5 Apply Properties of Logarithms 509

4 7.5 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 11, 17, and 71 5 TAKS PRACTICE AND REASONING Es. 43, 44, 64, 71, 73, 75, and VOCABULARY Copy and complete: To condense the epression log log 3 y, you need to use the? property of logarithms. 2. WRITING Describe two ways to evaluate log 7 12 using a calculator. EXAMPLE 1 on p. 507 for Es MATCHING EXPRESSIONS Match the epression with the logarithm that has the same value. 3. ln 6 2 ln ln ln 2 6. ln 6 1 ln 2 A. ln 64 B. ln 3 C. ln 12 D. ln 36 APPROXIMATING EXPRESSIONS Use log 4 ø and log 12 ø to evaluate the logarithm. 7. log 3 8. log log log log log 1 } log 1 } log 1 } 12 EXAMPLE 2 on p. 508 for Es EXPANDING EXPRESSIONS Epand the epression. 15. log ln log log log 2 2 } ln 12 } log 4 } 3y 22. ln 4 2 y 23. log yz log ln 2 y 1/3 26. log log 2 Ï } 28. ln 6 2 } y ln 4 Ï } log 3 Ï } 9 ERROR ANALYSIS Describe and correct the error in epanding the logarithmic epression. 31. log (log 2 5)(log 2 ) 32. ln ln 8 1 ln EXAMPLE 3 on p. 508 for Es CONDENSING EXPRESSIONS Condense the epression. 33. log log ln 12 2 ln log 1 log ln 1 4 ln y log 2 4 log y log log log 4 y 39. ln ln 1 } 2 1 ln 40. log } 3 log ln ln y 42. 2(log log 3 4) log MULTIPLE TAKS REASONING CHOICE Which of the following is equivalent to 3 log 4 6? A log 4 18 B log 4 72 C log D log Chapter 7 Eponential and Logarithmic Functions

5 44. MULTIPLE TAKS REASONING CHOICE Which of the following statements is not correct? A log log log 3 3 B log log log 3 6 C log log log 3 3 D log log log 3 3 EXAMPLE 4 on p. 509 for Es CHANGE-OF-BASE FORMULA Use the change-of-base formula to evaluate the logarithm. 45. log log log log log log log log log log log log log 6 24 } log 2 15 } log 3 9 } log 7 3 } ERROR ANALYSIS Describe and correct the error in using the change-of-base formula. log log 3 } log 7 EXAMPLE 5 on p. 509 for Es SOUND INTENSITY In Eercises 62 and 63, use the function in Eample Find the decibel level of the sound made by each object shown below. a. b. c. Barking dog: I W/m 2 Ambulance siren: I W/m 2 Bee: I W/m The intensity of the sound of a trumpet is 10 3 watts per square meter. Find the decibel level of a trumpet. 64. OPEN-ENDED TAKS REASONING MATH For each statement, find positive numbers M, N, and b (with b Þ 1) that show the statement is false in general. a. log b (M 1 N) 5 log b M 1 log b N b. log b (M 2 N) 5 log b M 2 log b N CHALLENGE In Eercises 65 68, use the given hint and properties of eponents to prove the property of logarithms. 65. Product property log b mn 5 log b m 1 log b n (Hint: Let 5 log b m and let y 5 log b n. Then m 5 b and n 5 b y.) 66. Quotient property log b m } n 5 log b m 2 log b n (Hint: Let 5 log b m and let y 5 log b n. Then m 5 b and n 5 b y.) 67. Power property log b m n 5 n log b m (Hint: Let 5 log b m. Then m 5 b and m n 5 b n.) 68. Change-of-base formula log c a 5 log b a } logb c (Hint: Let 5 log b a, y 5 log b c, and z 5 log c a. Then a 5 b, c 5 b y, and a 5 c z, so that b 5 c z.) 7.5 Apply Properties of Logarithms 511

6 PROBLEM SOLVING EXAMPLE 5 on p. 509 for Es CONVERSATION Three groups of people are having separate conversations in a room. The sound of each conversation has an intensity of watts per square meter. What is the decibel level of the combined conversations in the room? 70. PARKING GARAGE The sound made by each of five cars in a parking garage has an intensity of watts per square meter. What is the decibel level of the sound made by all five cars in the parking garage? 71. SHORT TAKS REASONING RESPONSE The intensity of the sound TV ads make is ten times as great as the intensity for an average TV show. How many decibels louder is a TV ad? Justify your answer using properties of logarithms. 72. BIOLOGY The loudest animal on Earth is the blue whale. It can produce a sound with an intensity of watts per square meter. The loudest sound a human can make has an intensity of watts per square meter. Compare the decibel levels of the sounds made by a blue whale and a human. 73. EXTENDED TAKS REASONING RESPONSE The f-stops on a 35 millimeter camera control the amount of light that enters the camera. Let s be a measure of the amount of light that strikes the film and let f be the f-stop. Then s and f are related by the equation: s 5 log 2 f 2 a. Use Properties Epand the epression for s. b. Calculate The table shows the first eight f-stops on a 35 millimeter camera. Copy and complete the table. Describe the pattern you observe. f s???????? c. Reasoning Many 35 millimeter cameras have nine f-stops. What do you think the ninth f-stop is? Eplain your reasoning. 5 WORKED-OUT SOLUTIONS 512 Chapter 7 Eponential p. WS1 and Logarithmic Functions 5 TAKS PRACTICE AND REASONING

7 74. CHALLENGE Under certain conditions, the wind speed s (in knots) at an altitude of h meters above a grassy plain can be modeled by this function: s(h) 5 2 ln (100h) a. By what factor does the wind speed increase when the altitude doubles? b. Show that the given function can be written in terms of common logarithms as s(h) 5 2 } log e (log h 1 2). MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Skills Review Handbook p. 1002; TAKS Workbook REVIEW Lesson 2.1; TAKS Workbook 75. TAKS PRACTICE Which of the following is not an eample of a Pythagorean triple? TAKS Obj. 10 A 8, 15, 17 B 48, 64, 80 C 7, 23, 25 D 11, 60, TAKS PRACTICE Which inequality best describes the range of the function whose graph is shown? TAKS Obj. 2 F y 21 G y 3 H y 21 J y y QUIZ for Lessons Evaluate the logarithm without using a calculator. (p. 499) 1. log log log log 1/2 32 Graph the function. State the domain and range. (p. 499) 5. y 5 log 2 6. y 5 ln y 5 log 3 ( 1 4) 2 1 Epand the epression. (p. 507) 8. log log ln 5y 3 6y log 3 } 8 Condense the epression. (p. 507) 12. log log ln 6 1 ln log log ln 2 5 ln Use the change-of-base formula to evaluate the logarithm. (p. 507) 16. log log log log SOUND INTENSITY The sound of an alarm clock has an intensity of I watts per square meter. Use the model L(I) 5 10 log I } I0, where I watts per square meter, to find the alarm clock s loudness L(I). (p. 507) EXTRA PRACTICE for Lesson 7.5, p ONLINE 7.5 Apply QUIZ Properties at classzone.com of Logarithms 513

8 Graphing Calculator ACTIVITY Use after Lesson Graph Logarithmic Functions TEKS a.5, a.6, 2A.11.B TEXAS classzone.com Keystrokes QUESTION How can you graph logarithmic functions on a graphing calculator? You can use a graphing calculator to graph logarithmic functions simply by using the or key. To graph a logarithmic function having a base other than 10 or e, you need to use the change-of-base formula to rewrite the function in terms of common or natural logarithms. E XAMPLE Graph logarithmic functions Use a graphing calculator to graph y 5 log 2 and y 5 log 2 ( 2 3) 1 1. STEP 1 Rewrite functions Use the change-of-base formula to rewrite each function in terms of common logarithms. y 5 log 2 y 5 log 2 ( 2 3) 1 1 5} log log 2 log ( 2 3) 5} 1 1 log 2 STEP 2 Enter functions Enter each function into a graphing calculator. STEP 3 Graph functions Graph the functions. Y1=log(X)/log(2) Y2=(log(X-3)/ log(2))+1 Y3= Y4= Y5= Y6= P RACTICE Use a graphing calculator to graph the function. 1. y 5 log 4 2. y 5 log 8 3. f() 5 log 3 4. y 5 log 5 5. y 5 log g() 5 log 9 7. y 5 log 3 ( 1 2) 8. y 5 log f() 5 log 4 ( 2 5) y 5 log 2 ( 1 4) y 5 log 7 ( 2 5) g() 5 log 3 ( 1 6) REASONING Graph y 5 ln. If your calculator did not have a natural logarithm key, eplain how you could graph y 5 ln using the key. 514 Chapter 7 Eponential and Logarithmic Functions