Simplify each expression. 1. log log log log log log x 3 - x 5. x 5 x 2 6. x6 1.

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1 8-. Plan Objectives To use the properties of logarithms Examples Identifying the Properties of Logarithms Simplifying Logarithms Expanding Logarithms Real-World Connection 8- What You ll Learn To use the properties of logarithms...and Why To relate sound intensity and decibel level, as in Example Properties of Logarithms Check Skills You ll Need GO for Help Lessons 8- and - Simplify each expression.. log + log 8 5. log 9 - log. log 6 log 6 Evaluate each expression for x =.. x - x 5. x 5 x 6. x6 8. x + x 6 x 9 Math Background Using the Properties of Logarithms Many students want to believe that log b (M + N) = log b M + log b N. However, this would be equivalent to adding the exponents in expressions such as x + x, which cannot be done. Thus, it is reasonable that there is no property for the logarithm of a sum. Do not confuse this with the valid property for a logarithm of a product, log b (MN) = log b M + log b N. More Math Background: p. 8C Lesson Planning and Resources See p. 8E for a list of the resources that support this lesson.. 0, 0.0, 0., 0.60, 0.699, 0.8, 0.85, 0.90, 0.95,,.6,.0. The sum of the logarithms equals the log of the product. Activity: Properties of Logarithms. Complete the table. Round to the nearest thousandth. x log x Complete each pair of statements. What do you notice? See left. a. log + log 5 = j and log (? 5) = j.6,.6 b. log + log = j and log (? ) = j 0.85, 0.85 c. log + log = j and log (? ) = j 0.90, 0.90 d. log 0 + log = j and log (0? ) = j.0,.0. Complete the statement: log M + log N = j. log (MN) M. a. Make a Conjecture How could you rewrite the expression log N M using the expressions log M and log N? log N log M log N b. Use your calculator to verify your conjecture for several values of M and N. Check students work. Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Logarithmic Functions as Inverses Lesson 8-: Example Extra Skills and Word Problems Practice, Ch. 8 Algebraic Expressions Lesson -: Examples, Extra Skills and Word Problems Practice, Ch. The properties of logarithms are summarized below. Key Concepts Properties Properties of Logarithms For any positive numbers, M, N, and b, b, log b MN = log b M + log b N Product Property log M b N = log b M - log b N Quotient Property log b M x = x log b M Power Property 5 Chapter 8 Exponential and Logarithmic Functions 5 Special Needs L Students who wear hearing aids, or who have difficulty hearing, may be sensitive about a discussion of noise levels and decibels. However, if these students are comfortable with the topic, invite them to contribute some of their special knowledge. Below Level L Have students state the Properties of Logarithms in words. Give illustrations using those words. Discuss whether these agree with the Property chosen.

2 nline Visit: PHSchool.com Web Code: age-05 a. log ± log b b. log y log c. log a ± log b You can use the properties of logarithms to rewrite logarithmic expressions. Identifying the Properties of Logarithms State the property or properties used to rewrite each expression. a. log 8 - log = log 8 Quotient Property: log 8 - log = log = log b. log b x y = log b x + log b y Product Property: log b x y = log b x + log b y Power Property: log b x + log b y = log b x + log b y State the property or properties used to rewrite each expression. a. log 5 + log 5 6 = log 5 Product Property b. log b - log b = log b 8 Power Property, Quotient Property You can write the sum or difference of logarithms with the same base as a single logarithm. Simplifying Logarithms Write each logarithmic expression as a single logarithm. a. log 0 - log log 0 - log = log 0 Quotient Property = log 5 Simplify. b. log x + log y log x + log y = log x + log y Power Property = log Ax yb Product Property So log 0 - log = log 5, and log x + log y = log Ax y B. a. Write log + log - log 6 as a single logarithm. log b. Critical Thinking Can you write log 9 - log 6 9 as a single logarithm? Explain. No; they do not have the same base. You can sometimes write a single logarithm as a sum or difference of two or more logarithms.. Teach Guided Instruction Activity Teaching Tip Remind students that a logarithm is an exponent. Therefore, it seems reasonable that there are properties for operations with logarithms that are similar to the properties for operations with exponents. Math Tip Point out to students that the bases are the same within each expression in part (a) and within each expression in part (b). The properties for logarithms do not apply unless the bases are the same. Additional Examples State the property or properties used to rewrite each expression. a. log 6 = log + log Product Property b. log x b y = log b x - log b y Quotient Property and Power Property Write each expression as a single logarithm. a. log 6 - log 6 log or b. 6 log 5 x + log 5 y log 5 (x 6 y) Vocabulary Tip In mathematics, to expand means to show the full form of. Expand each logarithm. Expanding Logarithms a. log x 5 y b. log r = log 5 x - log 5 y Quotient Property = log + log r Product Property = log + log r Power Property Expand each logarithm. See above left. a. log b b. log Q y R c. log a b Lesson 8- Properties of Logarithms 55 Advanced Learners L Ask students to explain how the properties of Logarithms on page 5 are similar to and how they are different from the properties of exponents. English Language Learners ELL Emphasize the correct reading of a logarithmic expression. For log b, students should say log base b of. Also, clarify that the term log by itself is meaningless. You cannot say y = log just as you cannot say y = ". 55

3 Error Prevention Discuss with students the fact that writing their exponents and bases clearly will help them avoid errors. It is easy to misread and confuse x these smaller digits so that log 5 y x might be misread as log 5AyB. Additional Examples Expand each logarithm. t a. log u log t log u b. log p log ± log p Manufacturers of a vacuum cleaner want to reduce its sound intensity to 0% of the original intensity. By how many decibels would the loudness be reduced? about four decibels. Resources Daily Notetaking Guide 8- L Daily Notetaking Guide 8- Adapted Instruction L Closure Ask students to write a paragraph describing how to use one logarithm property to simplify logarithm expressions. You may wish to assign properties so that all three are covered. Real-World Connection The workers who direct planes at airports must wear ear protection.. about 6 decibels Logarithms are used to model sound. The intensity of a sound is a measure of the energy carried by the sound wave. The greater the intensity of a sound, the louder it seems. This apparent loudness L is measured in decibels. You can use the formula I L = 0 log I, where I is the 0 intensity of the sound in watts per square meter AW/m B. I 0 is the lowest-intensity sound that the average human ear can detect. Real-World Connection Noise Control A shipping company has started flying cargo planes out of the city airport. Residents in a nearby neighborhood have complained that the cargo planes are too loud. Suppose the shipping company hires you to design a way to reduce the intensity of the sound by half. By how many decibels would the loudness of the sound be decreased? Relate The reduced intensity is one half of the present intensity. Define Let I = present intensity. Let I = reduced intensity. Let L = present loudness. Let L = reduced loudness. Write I = 0.5 I L = L = L - L = 0 log I I0 0 log I I0 Find the decrease in loudness L L. = 0 log I I0 0 log 0.5I I Substitute I 0.5I. 0 = 0 log I I0 0 log I I0 Motorcycle Hum of refrigerator Threshold of hearing 0 log I I0 0 log Q0.5? I I0 R Loudness of Sounds 50dB 80dB 0dB 00 db 5dB = 0 log I I0 0Qlog 0.5 log I I0 R Product Property Jet engine = 0 log I I0 0 log log I I0 Distributive Property =-0 log 0.5 Combine like terms. <.0 Use a calculator. The decrease in loudness would be about three decibels. Vacuum cleaner Breathing Suppose the shipping company wants you to reduce the sound intensity to 5% of the original intensity. By how many decibels would the loudness be reduced? 0dB 56 Chapter 8 Exponential and Logarithmic Functions 56. Product Property. Quotient Property. Power Property. Power Property 5. Power Property, Quotient Property 6. Power Property. Power Property, Quotient Property 8. Power Property, Product Property 9. Power Property, Quotient Property 0. Power Property, Product Property

4 EXERCISES Practice and Problem Solving A GO Practice by Example for Help Example (page 55) 0. See margin p. 56. Example (page 55) Example (page 55) State the property or properties used to rewrite each expression.. log + log 5 = log 0. log - log 8 = log. log z = log z. log 6 " n x p 5 p n log 6 x 5. 8 log - log 8 = log 6.. log 5 - log = log 8. log w + log z = log w z Q 5 R 9. log m - log n = log m 0. log b 8 log b = log b 8 Write each logarithmic expression as a single logarithm.. log + log log. log 9 - log log. 5 log + log log 9. log 8 - log 6 + log log 5. log m - log n log m n 6. log 5 - k log log 5 k Expand each logarithm. For more exercises, see Extra Skill and Word Problem Practice. n 9 0. See margin. 9. log x y 5 0. log xyz. log 5!x. log m n -. log 5 s r. log (x) 5. log (x - ) 6. log a b. log x Å y 8. log 8 8 Ía5 9. log s! 0. log b x c t log " x 5 log x xy. log 6 5 log 6 x log 6 5x 8. log x log y log z log z. Practice Assignment Guide A B -8 C Challenge Test Prep 9-95 Mixed Review Homework To check students understanding of key skills and concepts, go over Exercises 5,, 56, 5,, 5. Diversity Exercise Some students may not know what a sound barrier along a highway looks like. Ask students to bring pictures, draw a sketch, or tell where one can be seen nearby. Discuss the fact that highway sound barriers are often built to prevent highway noise from affecting neighborhoods. B Example (page 56) Apply Your Skills. The coefficient is missing in log s; t log " s t log s (log t log s) log t log s.. One brand of ear plugs claims to block the sound of snoring as loud as db. A second brand claims to block snoring that is eight times as intense. If the claims are true, for how many more decibels is the second brand effective? 9dB. A sound barrier along a highway reduced the intensity of the noise reaching a community by 95%. By how many decibels was the noise reduced? db Use the properties of logarithms to evaluate each expression.. log - log 6. log - log 5. log + 5 log 6 6. log + log 00. log 6 + log log 8 log log - log 0. log 5 - log 5 5. log 9 log 9. Error Analysis Explain why the expansion below of log t Å s is incorrect. Then do the expansion correctly. See left. log t Å s 5 log s t = log t log s. Open-Ended Write log 50 as a sum or difference of two logarithms. Answers may vary. Sample: log 50 log 5 ± log 0. GPS Enrichment Pearson Education, Inc. All rights reserved. Guided Problem Solving Reteaching Practice Name Class Date Practice 8- For Exercises, use the formula L 0 log I I. 0. A sound has an intensity of W/m.What is the loudness of the sound in decibels? Use I W>m. Properties of Logarithms. Suppose you decrease the intensity of a sound by 5%. By how many decibels would the loudness be decreased? Assume that log N 0., log N 0.60, and log 5 N Use the properties of logarithms to evaluate each expression. Do not use a calculator.. log. log 6 5. log 6. log log 5 8. log 6 9. log 6 - log 0. log 60 5 Write each logarithmic expression as a single logarithm.. log 5 + log 5. log log 6 5. log + log - log 8. 5 log x - log x 5. log 60 - log + log x 6. log - log + log 6. log x - log y 8. log r + log s - log t 9. log x + log 5y 0. 5 log - log. log x + log x. log + log + log. (log - log ) - log. 5 log x + log x 5. log 6 - log log + log - log. log x - 5 log y 8. (log x - log y) Expand each logarithm. 9. log x + log y - log z 0. ( log t ). log 5 y - (log 5 r + log 5 t). log xyz. log x yz. log 6x y 5. log (x - ) 6. log rst. log 5x Å 5w y 8. log 5 5x -5 x y 9. log 0. log (xyz) k State the property or properties used to rewrite each expression.. log 6 - log = log. 6 log = log 6. log x = log + log x " x " 9. log x = log 5. log = log 6. log 0 - log x = log 0 x L L L L Lesson 8- Properties of Logarithms 5 9. log x ± 5 log y 0. log ± log x ± log y ± log z. log 5 ± log x. log ± log m log n. log 5 r log 5 s. log ± log x 5. log ± log (x ) 6. log a ± log b log c. log ± log x log y 5 8. ± log 8 ± log 8 a 9. log s ± log log t 0. log b x 5

5 . Assess & Reteach 58. True; log and log False; log log, not log. 60. True; it is an example of the Power Property since Lesson Quiz Write each expression as a single logarithm. State the property you used.. log - log log ; Quotient Property. log 5 + log log (5? ); Power Property and Product Property Expand each logarithm. a. log c b log c a log c b. log x log x Use the properties of logarithms to evaluate each expression. 5. log log log y y Alternative Assessment Have students work in small groups to prepare an informal proof of an expanded logarithm from Exercises 9 8. They should justify each step, using the Properties of Logarithms. Then ask each group to present their proof at the board, and encourage class discussion of the proof. 6. False; the two logs have different bases. Real-World. No; the expression (x ± ) is a sum, so it is not covered by the Product, Quotient, or Power properties.. log "x. "y log x z 5. log 6. log m x ny p "x "y. log b z 5 8. log "z " "x 5 Connection Decibel meters are used to measure sound levels. GO nline Homework Help Visit: PHSchool.com Web Code: age Chapter 8 Exponential and Logarithmic Functions 6. False; this is not an example of the Quotient Property. log(x ) u log x log. x 6. False; log b y log b x log b y. Assume that log N 0.60, log 5 N , and log 6 N 0.8. Use the properties of logarithms to evaluate each expression. Do not use your calculator.. log log log 6.0. log log log log log log log log log! Noise Control New components reduce the sound intensity of a certain model GPS of vacuum cleaner from 0 - W/m to W/m. By how many decibels do these new components reduce the vacuum cleaner s loudness? 5. Reasoning If log x = 5, what is the value of db x? Write true or false for each statement. Justify your answer. 58. log + log 8 = log 5 log 60. log 8 = log 6. log log = log 5 8 log x log 6. log (x - ) = 6. b x log log = log x b y b y 6. Alog xb 5 log x 65. log - log = log 66. log x + log(x + ) = log(x + x) 6. log + log = log log x - log y = log x 69. log b + log b = log b 8 y See margin p Construction Suppose you are the supervisor on a road construction job. Your team is blasting rock to make way for a roadbed. One explosion has an intensity of W/m.What is the loudness of the sound in decibels? (Use I 0 = 0 - W/m.) 0 db. Critical Thinking Can you expand log (x + )? Explain.. Writing Explain why log (5? ) log 5? log. Write each logarithmic expression as a single logarithm. 8. See left.. log + log x. Qlog x log x yr log x z 5. log- log + log 9 6. x log m + y log n - log p log z log. log b x 8. 5 log x log b y 5 log b z Expand each logarithm See back of book. 9. log Á 5 x 80. log 8. log n p r Å s Í x "y 8. log Í x Í x b 8. log 5 y 8. log 5 "z zw (x ) x! 85. log 86. log 8. log Ír 9 c (xy) z d Ä y s t m 6. False; the exponent on the left means log x, quantity squared, not the log of x. 65. False; log log log, not log See margin. See margin p True; log x ± log (x ± ) log x(x ± ), which equals log (x ± x). 6. False; the three logs have different bases.

6 C Challenge 88. v log b N b v N MN b u? b v b u±v log b MN u ± v log b MN log b M ± log b N 88. Let u = log b M, and let v = log b N. Prove the Product Property of Logarithms by completing the equations below. Statement u = log b M b u = M v = j b v = j MN = b u b v = b j log b MN = j Reason Given Rewrite in exponential form. Given Rewrite in exponential form. Apply the Product Property of Exponents. Take the logarithm of each side. log b MN = log b j + log b j Substitute log b M for u and log b N for v. 89. Let u = log b M. Prove the Power Property of logarithms. See margin. 90. Let u = log b M and v = log b N. Prove the Quotient Property of logarithms. See margin. Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. 8 Test-Taking Strategies, p. 8 Test-Taking Strategies with Transparencies Test Prep GO Multiple Choice Short Response Extended Response Mixed Review for Help Lesson 8- Lesson -5 Lesson 6-5 lesson quiz, PHSchool.com, Web Code: aga True; the power and quotient properties are used correctly. 9. Which statement is NOT correct? B A. log 5 =? log 5 B. log 6 =? log 8 C. log 5 =? log 5 D. log 8 0,000 =? log Which expression is equal to log 5 + log? G F. log 8 G. log 5 H. log 5 J. log Which expression is equal to log 5 x +? log 5 y -? log 5 z? D xy (xy) A. log 5 (-8xyz) B. -log 5 C. log xy 5 D. log z 5 9. log 5 0 <.0 and log 5 0 <.86. Find the value of log 5 Q R without using a calculator. Explain how you found the value. See margin. 95. Use the properties of logarithms to write log in four different ways. Name each property you use. See back of book. Write each equation in logarithmic form. log = 9. 5 = = log 9 log 5 5 log 8 Solve each equation. Check for extraneous solutions. 00. " y = 6 0. " 6 8, 8 x = 0 0.!w 5!w A polynomial equation with integer coefficients has the given roots. What additional roots can you identify? 0.!,!5 ", "5 0. -i,ii, i 05. i, - + i i, i 06.!, i 0. -!,! ", "08. -i +, i i ±, i ", i 69. True; the left side equals log b ( 8? ), which equals log b 8. z Lesson 8- Properties of Logarithms 59 z. The log of a product is equal to the sum of the logs. log (MN) log M ± log N. So log (5? ) log 0, log 5? log N (0.)(0.) 0., which is not equal to u log b M (given). b u M b (Rewrite in exponential form.). (b u ) x M x (Raise each side to x power.). b ux M x (Power Property of exponents) 5. log b b ux log b M x (Take the log of each side.) 6. ux log b M x (Simplify.). log b M x x? log b M (substitution) 90.. u log b M (given). b u M (Rewrite in exponential form.). v log b N (given). b v N (Rewrite in exponential form.) 5. M b u b u v N b v (Quotient Property of Exponents) M 6. log b log b b u v N (Take the log of each side.) M. log b N u v (Simplify.) M 8. log b N log b M log b N (substitution) 9. [] By the Quotient Property, log 5 ( ) 0 log 5 ( 0) N [] correct answer, without work shown 59

7 8-5 Exponential and Logarithmic Equations 8-5. Plan What You ll Learn To solve exponential equations To solve logarithmic equations...and Why To model animal populations, as in Example 5 Solving Exponential Equations Check Skills You ll Need GO for Help Lessons 8- and - Evaluate each logarithm.. log 9 8? log 9. log 0? log 9. log 6 log 8. Simplify 5. 5 New Vocabulary exponential equation logarithmic equation Change of Base Formula An equation of the form b cx = a, where the exponent includes a variable, is an exponential equation. If m and n are positive and m = n, then log m = log n. You can therefore solve an exponential equation by taking the logarithm of each side of the equation. Objectives To solve exponential equations To solve logarithmic equations Examples Solving an Exponential Equation Using the Change of Base Formula Solving an Exponential Equation by Changing Bases Solving an Exponential Equation by Graphing 5 Real-World Connection 6 Solving a Logarithmic Equation Using Logarithmic Properties to Solve an Equation Solve x = 0. Solving an Exponential Equation x = 0 log x = log 0 Take the common logarithm of each side. x log = log 0 Use the power property of logarithms. log 0 x = log Divide each side by log. < 0.5 Use a calculator. Math Background The Change of Base Formula allows you to rewrite any logarithm in terms of a logarithm to any desired base. More Math Background: p. 8D Check x = 0 (0.5) < < 0 Solve each equation. Round to the nearest ten-thousandth. Check your answers. a. x =.69 b. 6 x = c. x+ = Lesson Planning and Resources See p. 8E for a list of the resources that support this lesson. Solve 6 x = 500. Solving an Exponential Equation by Graphing Graph the equations y = 6 x and y = 500. Find the point of intersection. The solution is x <.008. Solve 6x = 86 by graphing. 0.6 Intersection X=.009 Y=500 Lesson 8-5 Exponential and Logarithmic Equations 6 Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Logarithmic Functions as Inverses Lesson 8-: Example Extra Skills and Word Problems Practice, Ch. 8 Rational Exponents Lesson -: Example Extra Skills and Word Problems Practice, Ch. Special Needs L Clarify for students that the goal in solving an exponential equation is the same as for any equation, to isolate the variable on one side of the equal sign. However, because the variable is an exponent, students must take the logarithm of both sides. Below Level L When solving an exponential equation by taking the logarithm of both sides, the Power Property of Logarithms is used to solve for the variable. Review the Power Property. 6

8 . Teach Guided Instruction Math Tip Point out that you can take the common logarithm (using base 0) of both sides of the equation no matter what base occurs in the equation. This means that you can use the feature of a calculator that finds the common logarithm. Error Prevention When you enter y = 6 x, be sure to use parentheses to enter it as 6^(x). Connection to Biology Although the mathematical model may indicate a population in the single digits, the reality of living creatures will not fit the mathematical model exactly. There is a minimum population level below which an endangered species will probably not be able to survive. GO for Help To review solving equations by tables, see Lesson 5-5. Real-World Connection The U.S. population of peninsular bighorn sheep was 0 in 9. By 999, only 5 remained. Solving an Exponential Equation by Tables Solve the equation (.5 x ) = 6 to the nearest hundredth. Enter y = (.5 x ) - 6. Use tabular zoom-in to find the sign change, as shown at the right. The solution is x <.. Solve 6x = 86 using tables. Compare your result with your solution in. 0.6 Real-World Connection Zoology Refer to the photo. Write an exponential equation to model the decline in the population. If the decay rate remains constant, in what year might only five peninsular bighorn sheep remain in the United States? Step Enter the data into your calculator. Let 0 represent the initial year, 9. Step Use the ExpReg feature to find the exponential function that fits the data. Step Graph the function and the line y = 5. Step Find the point of intersection. X Y X=.09 The solution is x, and 9 + = 09, so there may be only five peninsular bighorn sheep in 09. ExpReg y = a*b^x a = 0 b = Intersection X=.8 Y=5 The population of peninsular bighorn sheep in Mexico was approximately 600 in 9. By 999, about 00 remained. Determine the year by which only 00 peninsular bighorn sheep might remain in Mexico. 068 Solving Logarithmic Equations To evaluate a logarithm with any base, you can use the Change of Base Formula. Key Concepts Property Change of Base Formula For any positive numbers, M, b, and c, with b and c, log b M = log c M log c b 6 Chapter 8 Exponential and Logarithmic Functions 6 Advanced Learners L Have students research what form of radioactive dating is most useful for charcoal that is less than 50,000 years old. English Language Learners ELL In Exercise 5, have a student volunteer explain the meaning of toxic. Discuss the pronunciation of protactinium, with help from the chemistry teacher or a dictionary. learning style: visual

9 5 6 Using the Change of Base Formula Use the Change of Base Formula to evaluate log 5. Then convert log 5 to a logarithm in base. log 5 = Use the Change of Base Formula. <.650 Use a calculator. log 5 = log x Write an equation..650 < log x Substitute log 5 N.650. x <.650 Write in exponential form. < 5.5 Use a calculator. The expression log 5 is approximately equal to.650, or log a. Evaluate log 5 00 and convert it to a logarithm in base 8.., log 8 0 b. Critical Thinking Consider the equation.65 < log x from Example. How could you solve the equation without using the Change of Base Formula? Answers may vary. Sample: Use a calculator to raise to the.65 power. An equation that includes a logarithmic expression, such as log 5 = log x in Example 5, is called a logarithmic equation. log 5 log Solve log (x + ) = 5. Method log (x + ) = 5 x + = 0 5 Solving a Logarithmic Equation x + = 00,000 x =, Solve for x. Write in exponential form. Method Graph the equations y = log (x + ) and y = 5. Use Xmin = 0000, Xmax = 0000, Ymin =.9, Ymax = 5.. Find the point of intersection. The solution is x =,. Method Enter y = log (x + ) - 5. Use tabular zoom-in to find the sign change. Use the information from Methods or to help you with your TblSet values. The solution is x =,. Check log (x + ) = 5 log (?, + ) 0 5 log 00, log 0 5 = 5 6 Solve log ( - x) =-. Check your answer..5 Intersection X= Y=5 X Y=0 Y E 5 E 5 E E 5.6E 5.9E 5 Lesson 8-5 Exponential and Logarithmic Equations 6 Additional Examples Solve 5 x = 6. about 0.86 Solve x = 00 by graphing. x N.68 Solve 5 x = 0 using tables. about.9 The population of trout in a certain stretch of the Platte River is shown for five consecutive years in the table, where 0 represents the year 99. If the decay rate remains constant, in the beginning of which year might at most 00 trout remain in this stretch of river? 05 Time t 0 Pop. P(t) Use the Change of Base Formula to evaluate log 6. Then convert log 6 to a logarithm in base. about.8; about log.589 Guided Instruction Teaching Tip Ask a volunteer to explain why the logarithm of a negative number must be undefined. 6 Additional Examples Solve log (x - ) =. 500 Solve log x - log = 5. about 58.8 Resources Daily Notetaking Guide 8-5 L Daily Notetaking Guide 8-5 Adapted Instruction L Closure Ask students to give examples of equations that can be solved by using the properties of exponents and logarithms. 6

10 . Practice Assignment Guide GPS Enrichment Guided Problem Solving Reteaching Practice Practice 8-5 A B -, 8-5, 58, 60-6, 66, 6-8, 85, 8, 89-9, 9, 95, 96 A B 5-, 55-5, 59, 6, 65, 6-5, 8-8, 86, 88, 9, 9 C Challenge 9-05 Test Prep 06- Mixed Review - Homework To check students understanding of key skills and concepts, go over Exercises,, 9, 6,, 8. Name Class Date Exponential and Logarithmic Equations Use the Change of Base Formula to evaluate each expression. Round answers to the nearest hundredth.. log. log 0. log 8. log 5 5. log 6. log 5 0. log 8 8. log 6 9. log 9 0. log 8 Solve each equation. Check your answer. Round answers to the nearest hundredth.. x =. n =. 5 x = 0. 8 n+ = 5. n- = 6. n = 5. 5 n- = 5 8. x - 5 = Solve each equation. Check your answer. Round answers to the nearest hundredth. 9. log x = 0. log x =. log (x - ) =. log x - log 5 =-. log 8 - log x =-. log (x + ) + log x = 5. 8 log x = 6 6. log x =. log x = 8. log (x - 5) = 9. log x = 0. log x - log 5 = Use the Change of Base Formula to solve each equation. Round answers to the nearest hundredth.. 0 x = 8. 8 n =. 0 x = 9. 5 n+ = 5. 0 n- = n = n-5 = x - 50 = The function y 000(.005) x models the value of $000 deposit at 6% per year (0.005 per month) x months after the money is deposited. 9. Use a graph (on your graphing calculator) to predict how many months it will be until the account is worth $ Predict how many years it will be until the account is worth $5000. Solve each equation. Round answers to the nearest hundredth.. log x - log 9 =. log x - log =-. log x - log =-. log x - log = 5. log x - log = 6. log (x + 5) =. log (x + 5) = 8. log x =- 9. log x - log = L L L L Pearson Education, Inc. All rights reserved. EXERCISES Example (page 6) Example (page 6) Example (page 6) Example (page 6) Example 5 (page 6) Sometimes the properties of logarithms will help solve an equation. Practice and Problem Solving A GO Practice by Example for Help Solve log x - log =. log x - log = Using Logarithmic Properties to Solve an Equation log Q x R = Write as a single logarithm. x = 0 Write in exponential form. x = (00) Multiply each side by. x = 0!, or about. Log x is defined only for x 0, so the solution is 0!, or about.. Solve log 6 - log x =-. 00 For more exercises, see Extra Skill and Word Problem Practice. Solve each equation. Round to the nearest ten-thousandth. Check your answers.. x = x = x = x = x = x =-0. 9 y = z = 0 9. x = 6 0. y- = 0. 5 x+ =. x- = Solve by graphing.. x = x = x = x = Use a table to solve each equation. Round to the nearest hundredth.. x+ = x- = x = x- = x , x+ = x -0.. x- = x.. An investment of $000 earns 5.5% interest, which is compounded quarterly. After approximately how many years will the investment be worth $000? about. years. The equation y = 8(.0) x models the U.S. population y, in millions of people, x years after the year 000. Graph the function on your graphing calculator. Estimate when the U.S. population will reach 50 million. about 08 Use the Change of Base Formula to evaluate each expression. Then convert it to a logarithm in base log 9 5. See margin. 6. log 8. log 5 8. log log 0. log. log log.6 Solve by graphing. Round answers to the nearest hundredth n = 5. 0 y = k- = x = 5. x = x+5 = 5 Example 6 (page 6). 60, or N "0, or N 5. Solve each equation. Check your answers. "0. 0 or about 0.6. log x =-. log x =- 5. log (x + ) = log x + = 8. log 6x - =- 8. log (x - ) = 0,000 "0, or 9. log x =.5 0. log (x + ) = 5. log (5 - x) = 0 about.6 6 Chapter 8 Exponential and Logarithmic Functions ; log ; log ; log ; log ; log ; log 8..8; log ; log 8.90

11 B Real-World Example (page 6) Apply Your Skills 6a. Florida growth factor.0, y 5,98,8? (.0) x ; New York growth factor.005, y 8,96,5? (.005) x b. 0 Connection Careers Seismologists use models to determine the source, nature, and size of seismic events. 6a. Texas growth factor.008, y 0,85,80? (.008) x ; California growth factor.0, y,8,68? (.0) x 8a b Solve each equation.. log x - log = ,000"5, or about, log x + log x =. log x + log = 5 5. log 5 - log x = 6. log x - log 6 + log. = 9. log (x + ) = log (x - ) Consider the equation x = 80. a c. See margin. a. Solve the equation by taking the logarithm in base 0 of each side. b. Solve the equation by taking the logarithm in base of each side. c. Writing Compare your result in parts (a) and (b). What are the advantages of either method? Explain. 9. Seismology An earthquake of magnitude.9 occurred in 00 in Gujarat, GPS India. It was,600 times as strong as the greatest earthquake ever to hit Pennsylvania. Find the magnitude of the Pennsylvania earthquake. (Hint: Refer to the Richter Scale on page 6.) 5. Write an equation. Then solve the equation without graphing. 50. A parent raises a child s allowance by 0% each year. If the allowance is $8 now, when will it reach $0? 0 8(.) x, 5 years 5. Protactinium-m,a toxic radioactive metal with no known use, has a half-life of. minutes. How long does it take for a 0-mg sample to decay to mg? See margin. 5. Multiple Choice As a town gets smaller, the population of its high school decreases by % each year. The student body has 5 students now. In how many years will it have about 5 students? A years years 0 years years Mental Math Solve each equation. 5. x = 5. x = 55. log 9 = x 56. log 6 = x 5. log 8 = x x = 59. log = x x = 00 5 Population Use this Most Populous States table for Exercises 6 6. Rank in 000 State California Texas New York Florida Most Populous States 000 Population,8,68 0,85,80 8,96,5 5,98,8 Average Annual Percentage Increase Since 990.0%.08% 0.5%.% SOURCE: U.S. Census Bureau. Go to for a data update. Web Code: agg a. Determine the growth factors for Florida and New York. Then write an equation to model each state s population growth. a b. See left. b. Estimate when Florida s population might exceed New York s population. 6. a. Determine the growth factors for Texas and California. Then write an equation to model each state s population growth. 06 b. Estimate when Texas s population might exceed California s population. 6. Critical Thinking Is it likely that Florida s population will exceed that of Texas? Explain your reasoning. See margin. c. Answers may vary. Sample: You don t have to use the change of base formula with the base-0 method, but there are fewer steps with the base- method. Lesson 8-5 Exponential and Logarithmic Equations 65 x 5.. 0( ),. min Exercise 5 Show students that they can simply multiply the initial population of 5 by the multiplier - 0. = 0.88, and determine the number of multiplications needed to get around 5. Alternatively, students can write an exponential equation that represents the situation and solve it. Alternative Method Exercise 5 Organize students in groups of. Have each student in the group try a different method: making a table to show successive values, graphing, or solving without graphing. Then have them discuss the relative merits of each method. 6. Since Florida s growth rate is larger than Texas s growth rate, in theory, given constant conditions, Florida would exceed Texas in about 5 years. However, since no state has unlimited capacity for growth, it is unrealistic to predict over a long period of time. 65

12 Exercise 6 This exercise can make the Change of Base Formula seem less like an arbitrary rule, thereby making it easier to remember. Auditory Learners Exercise 9 Ask a volunteer to find a picture of piano strings (or actually take the class to see a piano if one is nearby) and discuss how the relative length of the strings relates to the pitch of each note on the keyboard. You could also use a guitar or a zither to demonstrate this. 65. Answers may vary. Sample: log x x, x N Answers may vary. Sample: A possible model is y 65(.088) x where x 0 represents 99; the growth is probably exponential and 65(.088) 0 N 6; using this model, there will be 0,000 manatees in about 05. log b log a b. x log a b 6a. x c. Substituting the result from part (a) into the results from part (b), or vice versa, yields log b log a b log a. This justifies the Change of Base Formula for c 0. a. top up: 0 5 W/m, top down: 0.5 W/m b % log b log a Real-World Connection Many Florida manatees die after collisions with motorboats. 5. GO log (x ) log x nline Homework Help Visit: PHSchool.com Web Code: age Error Analysis What is wrong with the proof below that =? log 0 log 0 = = log 0 = log 0 5 log 0 u log 0 5 log Open-Ended Write and solve a logarithmic equation. 66. Zoology Conservation efforts have increased the endangered Florida manatee population from 65 in 99 to 6 in 00. If this growth rate continues, when might there be 0,000 manatees? Explain the reasoning behind your choice of a model. See margin. 6. Consider the equation a x = b. a. Solve the equation by using log base 0. a c. See margin. b. Solve the equation by using log base a. c. Use your results in parts (a) and (b) to justify the Change of Base Formula. Write each logarithm as the quotient of two common logarithms. Do not simplify the quotient. log log 8 log 0 log. 68. log log 69. log 8 0. log 5 0. log 9. log log 5 log 9 log x log 5. log x. log 6 ( - x). log x 5 5. log x (x + ) log log ( x) log x See left. log 6 Acoustics In Exercises 6 8, the loudness measured in decibels (db) is defined by I loudness 0 log, where I is the intensity and I 0 0 W/m. I 0 See margin. 6. The human threshold for pain is 0 db. Instant perforation of the eardrum occurs at 60 db. a. Find the intensity of each sound. 0 0 (or ) W/m, 0 W/m b. How many times as intense is the noise that will perforate an eardrum as the noise that causes pain? 0,000 times as intense. The noise level inside a convertible driving along the freeway with its top up is 0 db. With the top down, the noise level is 95 db. a b. See margin. a. Find the intensity of the sound with the top up and with the top down. b. By what percent does leaving the top up reduce the intensity of the sound? 8. A screaming child can reach 90 db. A launch of the space shuttle produces sound of 80 db at the launch pad. a. Find the intensity of each sound. 0 W/m, 0 6 W/m b. How many times as intense as the noise from a screaming child is the noise from a shuttle launch? 0 9 times more intense Solve each equation. If necessary, round to the nearest ten-thousandth x = x = x = log x + log = log - log x = log x = x = log 8 (x - ) = x = log (5x - ) = x = x = x = 5 9. log + log x = x = log x =. 95. x + 0. = x - =.0 66 Chapter 8 Exponential and Logarithmic Functions 66

13 C Challenge 9a. bassoon, guitar, harp, violin, viola, cello b. bassoon, guitar, harp, cello, bass c. harp, violin d. harp, violin 0. 0,0 m above sea level 05b mg or.06 mg c. Estimate in hours is more accurate; the days have a larger rounding error. Bass Bassoon Guitar Cello Harp Violin Viola Music The pitch, or frequency, of a piano nnote is related to its position on the keyboard by the function F(n) = 0?,where F is the frequency of the sound wave in cycles per second and n is the number of piano keys above or below Concert A, as shown above. If n = 0 at Concert A, which of the instruments shown in the diagram can sound notes of the given frequency? a. 590 b. 0 c. 0 d Astronomy The brightness of an astronomical object is called its magnitude. A decrease of five magnitudes increases the brightness exactly 00 times. The sun is magnitude -6., and the full moon is magnitude -.5. The sun is about how many times brighter than the moon? 8,60 times 99. Archaeology A scientist carbon-dates a piece of fossilized tree trunk that is thought to be over 5000 years old. The scientist determines that the sample contains 65% of the original amount of carbon-. The half-life of carbon- is 50 years. Is the reputed age of the tree correct? Explain. No; solving (0.5) 50 x for x, the age in years of the sample, Solve each equation. yields an age of about 56 yr. 00. log (x - ) = 5 0. log (x + x) =, 0. log (x - ) = 9, 9 0. log - log x = 0. In the formula P = P 0 A B 95 h, P is the atmospheric pressure in millimeters of mercury at elevation h meters above sea level. P 0 is the atmospheric pressure at sea level. If P 0 equals 60 mm, at what elevation is the pressure mm? 05. Chemistry A technician found mg of a radon isotope in a soil sample. After hours, another measurement revealed 0 mg of the isotope. a. Estimate the length of the isotope s half-life to the nearest hour and to the nearest day. 9 hours or days b. For each estimate, determine the amount of the isotope after two weeks. c. Compare your answers to part (b). Which is more accurate? Explain.. Assess & Reteach Lesson Quiz Use mental math to solve each equation.. x = 8. log = x. 0 6x = 0. Solve 5 x = 5. Alternative Assessment Have a class discussion. Ask students to talk about which aspect of solving exponential and logarithmic equations they found most confusing or difficult, citing specific examples from the lesson. Have other students give tips and explanations to help clarify the topics. Be sure each student contributes to the discussion. Test Prep A sheet of blank grids is available in the Test-Taking Strategies with Transparencies booklet. Give this sheet to students for practice with filling in the grids. Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. 8 Test-Taking Strategies, p. 8 Test-Taking Strategies with Transparencies Test Prep Gridded Response Use a calculator to solve each equation. Enter each answer to the nearest hundredth x = x-5 = = 5 x- lesson quiz, PHSchool.com, Web Code: aga-0805 Lesson 8-5 Exponential and Logarithmic Equations 6 6

14 Use this Checkpoint Quiz to check students understanding of the skills and concepts of Lessons 8- through 8-5. Resources Grab & Go Checkpoint Quiz Gridded Response Mixed Review Use the Change of Base Formula to solve each equation. Enter the answer to the nearest tenth. 09. log 5 x = log log 9 x = log 6 5. Solve each equation.. log ( + x) =. log (x - ) = 0 GO for Help Lesson 8- Expand each logarithm. 8. See margin.. log x y -. log x y 5. log (x) 6. log (x - ). log 5!x 8. log Q 5a b R Lesson -6 Let ƒ(x) = x and g(x) = x. Perform each function operation.. log ± log x log y. log x log y 5. log ± log x 6. log ± log (x ). log 5 ± log x 8. log 5 ± log a log b Checkpoint Quiz. y Lesson 6-6 Lesson - 9. (ƒ + g)(x) 0. (g - ƒ)(x). (ƒ? g)(x) x ± x x x x x Find all the zeros of each function.. y = x - x + x -, w i. ƒ(x) = x - 6 w, wi. ƒ(x) = x - 5x + 6w", w" 5. y = x - x - 8,, Write an equation to solve each problem. 6. A customer at a hardware store mentions that he is buying fencing for a vegetable garden that is ft longer than it is wide. He buys 8 ft of fencing. What is the width of the garden? (x) ± (x ± ) 8; 6 ft. A bowler has an average of. In a set of games one night, her scores are 5,, 9,, and 56. What score must she bowl in the sixth game to maintain her average? 69 x 6 ; 9. O O y 9 8. log 6 ± log 6 x ± log 6 y 5. log 6 ± log 6 x log 0 0. Rewrite log 0 as log and evaluate it to get N.. Then set. log x. Rewrite to get log x. log and solve. Convert log x.585 to x or x N 8.6. So log 0 N log 8.6. x x Checkpoint Quiz Lessons 8- through log 0.00, log 0.609, log.5850, 8, 9 68 Chapter 8 Exponential and Logarithmic Functions Graph each logarithmic function.. See margin.. y = log 6 x. y = log (x - ) Expand each logarithm. 5. See margin.. log s log s 5 log r. log 6 (xy) 5. log 6!x r 5 Solve each equation. 6. x 5. log 5x = 0 8. log x = Evaluate the expressions below and order them from least to greatest. log log log 0. Writing Explain how to use the Change of Base Formula to rewrite log 0 as a logarithmic expression with base. See margin. 68

15 8-6. Plan 8-6 Natural Logarithms Objectives To evaluate natural logarithmic expressions To solve equations using natural logarithms Examples Simplifying Natural Logarithms Real-World Connection Solving a Natural Logarithmic Equation Solving an Exponential Equation 5 Real-World Connection Math Background Natural logarithms, or logarithms with base e, occur in many problems involving the growth and decay of natural organisms. In fact, natural logarithms arise in those problems which model continuous growth and decay as discussed in Lesson 8-. More Math Background: p. 8D What You ll Learn To evaluate natural logarithmic expressions To solve equations using natural logarithms...and Why To model the velocity of a rocket, as in Example Natural Logarithms Check Skills You ll Need GO for Help Lessons 8- and 8-5 Use your calculator to evaluate each expression to the nearest thousandth.. e 5. e. e e e. Solve. 6. log x = 8. log 6 = x 8. log 6 x = 65,56 New Vocabulary natural logarithmic function In Lesson 8-, you learned that the number e <.88 can be used as a base for exponents. The function y = e x has an inverse, the natural logarithmic function. Key Concepts Definition Natural Logarithmic Function If y = e x, then log e y = x, which is commonly written as ln y = x. The natural logarithmic function is the inverse, written as y = ln x. ➀ ➁ ➀ y e x ➁ y ln x Lesson Planning and Resources See p. 8E for a list of the resources that support this lesson. 0 Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Properties of Exponential Functions Lesson 8-: Example Extra Skills and Word Problems Practice, Ch. 8 Exponential and Logarithmic Equations Lesson 8-5: Example 6 Extra Skills and Word Problems Practice, Ch. 8 Vocabulary Tip ln y means the natural logarithm of y. The l stands for logarithm and the n stands for natural. 0 Chapter 8 Exponential and Logarithmic Functions The properties of common logarithms apply to natural logarithms also. Special Needs L In Example, students may fail to take the natural logarithm of both sides. Stress that if any logarithm is taken on one side of the equation, the same logarithm must be applied to the other side. Ask: Why was the natural logarithm used? base is e Simplifying Natural Logarithms Write ln 6 - ln 8 as a single natural logarithm. ln 6 - ln 8 = ln 6 - ln 8 Power Property = ln 6 8 Quotient Property = ln Simplify. Write each expression as a single natural logarithm. a. 5 ln - ln ln 8 b. ln x + ln y ln x y c. ln ln x ln "x Below Level L Review the general formula for growth or decay, N = N 0 e kt and the formula for continuously compounded interest, A = Pe rt. Compare the meanings of corresponding variables.

16 Real-World Connection The space shuttle is launched into orbit. Natural logarithms are useful because they help express many relationships in the physical world. Real-World Connection Space A spacecraft can attain a stable orbit 00 km above Earth if it reaches a velocity of. km/s. The formula for a rocket s maximum velocity v in kilometers per second is v = t + c ln R.The booster rocket fires for t seconds and the velocity of the exhaust is c km/s. The ratio of the mass of the rocket filled with fuel to its mass without fuel is R. Suppose a rocket used to propel a spacecraft has a mass ratio of 5, an exhaust velocity of.8 km/s, and a firing time of 00 s. Can the spacecraft attain a stable orbit 00 km above Earth? Let R = 5, c =.8, and t = 00. Find v. v = t + c ln R Use the formula. = (00) +.8 ln 5 Substitute. < (.9) Use a calculator. < 8.0 Simplify. The maximum velocity of 8.0 km/s is greater than the. km/s needed for a stable orbit. Therefore, the spacecraft can attain a stable orbit 00 km above Earth. N5. km/s; no a. A booster rocket for a spacecraft has a mass ratio of about 5, an exhaust velocity of. km/s, and a firing time of 0 s. Find the maximum velocity of the spacecraft. Can the spacecraft achieve a stable orbit 00 km above Earth? b. Critical Thinking Suppose a rocket, as designed, cannot provide enough velocity to achieve a stable orbit. Look at the variables in the velocity formula. What alterations could be made to the rocket so that a stable orbit could be achieved? One could increase its mass ratio or its exhaust velocity.. Teach Guided Instruction Teaching Tip Point out that the speed is in kilometers per second. To put this in more familiar units so students can get a sense of how fast the rocket is moving, suggest that they convert this speed to km/h or mi/h. Additional Examples Write ln - ln 9 as a single natural logarithm. ln 6 Find the velocity of a spacecraft whose booster rocket has a mass ratio of, an exhaust velocity of. km/s, and a firing time of 50 s. Can the spacecraft achieve a stable orbit 00 km above Earth? about 6.6 km/s; no Natural Logarithmic and Exponential Equations Guided Instruction You can use the properties of logarithms to solve natural logarithmic equations. Solve ln (x + 5) =. Solving a Natural Logarithmic Equation ln (x + 5) = (x + 5) = e Rewrite in exponential form. (x + 5) < 5.60 Use a calculator. x + 5 < "5.60 Take the square root of each side. x + 5 <.9 or -.9 Use a calculator. x < 0.9 or -.0 Solve for x. Error Prevention Remind students that the parentheses in ln (x + 5) are necessary in order to show that you are taking the natural logarithm of the entire binomial. Check ln (? ) 0 ln (? (-.0) + 5) 0 ln ln <.000 < Solve each equation. Check your answers. a. ln x = b. ln (x - 9) = c. ln Q x R = 9,605,.8 88,6. Lesson 8-6 Natural Logarithms Advanced Learners L Ask students to solve? x + = by taking logs to the base and using the change of base formula. English Language Learners ELL Help students make the connection between natural logarithms and the math term ln. Discuss the pronunciation of ln as the letters sound, or ell n. Write ln x = log e x on the board and have students read it to emphasize the meaning of natural logarithms.

17 Additional Examples Solve ln (x - ) = 6. about Use natural logarithms to solve e x +. =. about An initial investment of $00 is now valued at $5.5. The interest rate is 6%, compounded continuously. How long has the money been invested? about years Technology Tip Have students check to see if their calculator has separate keys for LOG and LN. Resources Daily Notetaking Guide 8-6 L Daily Notetaking Guide 8-6 Adapted Instruction L Closure Ask students to compare and contrast natural and common logarithms. Answers may vary. Sample: They are both exponents; they both obey the same properties of exponents and logarithms; they use a different base. The base for common logarithms is the rational number 0; for natural logarithms, it is the irrational number e. A B C D E A B C D E A B C D E A B C D E 5 A B C D E B C D E Test-Taking Tip If you perform an operation on one side of an equation, remember to perform the same operation on the other side. EXERCISES 5 You can use natural logarithms to solve exponential equations. Solving an Exponential Equation Use natural logarithms to solve e x +.5 = 0. e x +.5 = 0 Practice and Problem Solving e x =.5 Subtract.5 from each side. e x =.5 Divide each side by. ln e x = ln.5 x = ln.5 Take the natural logarithm of each side. Simplify. x = ln.5 Solve for x. x < 0.58 Use a calculator. Use natural logarithms to solve each equation. a. e x+ = 0.0 b. e x 5. = Real-World Connection Multiple Choice An investment of $00 is now valued at $9.8. The interest rate is 8%, compounded continuously. About how long has the money been invested? years 5 years years 9 years A = Pe rt Continuously compounded interest formula 9.8 = 00e 0.08t Substitute 9.8 for A, 00 for P, and 0.08 for r..98 = e 0.08t Divide each side by 00. ln.98 = ln e 0.08t Take the natural logarithm of each side. ln.98 = 0.08t Simplify. ln = t Solve for t. 5 < t Use a calculator. The money has been invested for about five years. The answer is B. 5 An initial investment of $00 is worth $5. after seven years of continuous compounding. Find the interest rate. 6.5% For more exercises, see Extra Skill and Word Problem Practice. A GO Practice by Example for Help Example (page 0) Example (page ) Write each expression as a single natural logarithm.. ln5 ln 5. ln 9 + ln ln 8. ln - ln 6 ln. ln 8 + ln 0 ln 0, ln - 5 ln ln 8 6. ln 8 - ln ln. 5 ln m - ln n 8. (ln x + ln y) - ln z 9. ln a - ln b + ln m5 ln c n "xy ln a"c Find the value of y for the given value of x. z ln b 0. y = 5 + ln x, for x = y = ln x, for x = Chapter 8 Exponential and Logarithmic Functions

18 GO B Example (page ) Example (page ) Example 5 (page ) Apply Your Skills for Help For Gridded Responses, see Test-Taking Strategies, page 6. For Exercises and, use v t ± c ln R.. Space Find the velocity of a spacecraft whose booster rocket has a mass ratio of 0, an exhaust velocity of. km/s, and a firing time of 0 s. Can the spacecraft achieve a stable orbit 00 km above Earth?.9 km/s; yes. A rocket has a mass ratio of and an exhaust velocity of.5 km/s. Determine the minimum firing time for a stable orbit 00 km above Earth. 5 s Solve each equation. Check your answers ln x = ln x = ln (x - l) = 6. ln (m + ) = 8 8. ln (t - ) = 5.8, ln x = 6 0. ln x. ln r.8 =. ln x w = 0.96 w. w0.908 Use natural logarithms to solve each equation.. e x = e x = e x + = = 5.9. e x = = 6 e x 5 9. Investing An initial deposit of $00 is now worth $.0. The account earns 8.% interest, compounded continuously. Determine how long the money has been in the account. 6 years 0. An investor sold 00 shares of stock valued at $.50 per share. The stock was purchased at $.5 per share two years ago. Find the rate of continuously compounded interest that would be necessary in a banking account for the investor to make the same profit. 8% Mental Math Simplify each expression.. ln e. ln e. ln e ln e 0 ln 5. ln 0 6. e. 8. ln e Gridded Response The battery power available to run a satellite is given by the formula P = 50 e t 50, where P is power in watts and t is time in days. For how many days can the satellite run if it requires 5 watts? 0 0. Space Use the formula for maximum velocity v = t + c ln R.Find the GPS mass ratio of a rocket with an exhaust velocity of. km/s, a firing time of 50 s, and a maximum shuttle velocity of 6.9 km/s. 0.8 Determine whether each statement is always true, sometimes true, or never true.. ln e x. ln e x = ln e x +. ln t = log e t sometimes never always Biology For Exercises 6, use the formula H Q r R (ln P ln A). H is the number of hours, r is the rate of decline, P is the initial bacteria population, and A is the reduced bacteria population. about 5.8% per hour. A scientist determines that an antibiotic reduces a population of 0,000 bacteria to 5000 in hours. Find the rate of decline caused by the antibiotic. 5. A laboratory assistant tests an antibiotic that causes a rate of decline of 0.. How long should it take for a population of 8000 bacteria to shrink to 500? about 9.8 h 6. A scientist spilled coffee on the lab report shown at the left. Determine the initial population of the bacteria. about 0,000 bacteria ln e e x 9.5 GPS Enrichment Pearson Education, Inc. All rights reserved.. Practice Assignment Guide Guided Problem Solving Reteaching Practice A B -, -8 A B -0, 9-6 C Challenge 6-66 Test Prep 6-0 Mixed Review -80 Homework To check students understanding of key skills and concepts, go over Exercises 8, 8, 0, 9,, 56. Alternative Method Exercises 8 Suggest that students ask themselves the question in a different form. For example, for Exercise ask: What power of the base e gives me the number e? Name Class Date Practice 8-6 The formula P 5 50e 5 t gives the power output P, in watts, available to run a certain satellite for t days. Find how long a satellite with the given power output will operate. Round answers to the nearest hundredth.. 0 W. W. W Natural Logarithms The formula for the maximum velocity v of a rocket is v c ln R, where c is the velocity of the exhaust in km/s and R is the mass ratio of the rocket.a rocket must reach.8 km/s to attain a stable orbit. Round answers to the nearest hundredth.. Find the maximum velocity of a rocket with a mass ratio of about 8 and an exhaust velocity of. km/s. Can this rocket achieve a stable orbit? 5. What mass ratio would be needed to achieve a stable orbit for a rocket with an exhaust velocity of.5 km/s? 6. A rocket with an exhaust velocity of. km/s can reach a maximum velocity of.8 km/s. What is the mass ratio of the rocket? Use natural logarithms to solve each equation. Round answers to the nearest hundredth.. e x = 5 8. e x = 0 9. e x+ = e x- = 5. e x- =. 5e 6x+ = 0.. e x =. e x e x-5 = 9 6. e 5x+8 = e x = e x e x = 5 0. e ln5x = 0. e ln x =. e x = Solve each equation. Check your answer. Round answers to the nearest hundredth.. ln x =-. ln (x - ) = 5. 5 ln (x - 6) = ln x =. - ln (8x + ) = 8. ln x + ln x = 9. ln x + ln x = 0. ln x + ln =. ln x - ln 5 =-. ln e x =. ln e x =. ln e x+5 = 5. ln x + ln x = 6. 5 ln (x - ) = 5. ln (x + 5) = 8 8. ln (x + ) = 5 9. ln x = 0. ln (x - ) = Write each expression as a single natural logarithm.. ln 6 - ln 8. ln + ln 9. a ln - ln b. ln z - ln x 5. ln 9 ln x 6. ln x + ln y L L L L lesson quiz, PHSchool.com, Web Code: aga-0806 Lesson 8-6 Natural Logarithms

19 . Assess & Reteach Lesson Quiz. Write ln 6 - ln as a single natural logarithm. ln. Solve e x = 5. about Simplify ln e. Alternative Assessment Ask students to discuss with a partner the differences in the procedures and answers for solving these two equations: ln x - ln (x -) = and log x - log (x -) =. Then have each pair write a paragraph that summarizes their discussion. The procedures are the same. If b is the base for the logarithm, then the answer for both equations is b (b ). However, for the first equation, b e and for the second equation, b 0. The answers are about.5 and about.00. GO nline Homework Help Visit: PHSchool.com Web Code: age no solution C Challenge Savings Suppose you invest $500 at 5% interest compounded continuously. Copy and complete the table to find how long it will take to reach each amount Amount (A) $600 $00 $800 $900 $000 $00 $00 $00 Time (years) Solve each equation ln x - ln= 56. ln (x - ) = 0 5. e x+ = ln (5x ) e x- + = e x- = e x + 6. ln x ln ln 5,.9 6. ln (x + ) - ln = Critical Thinking Can ln 5 + log 0 be written as a single logarithm? Explain. See margin. 6. In 000, there were about 00 million Internet users. That number is projected to grow to billion in 005. y 00e 0.t a. Let t represent the time, in years, since 000. Write a function of the form y = ae ct that models the expected growth in the population of Internet users. b. In what year might there be 500 million Internet users? 00 c. In what year might there be.5 billion Internet users? 006 d. Solve your equation for t. d e. See margin. e. Writing Explain how you can use your equation from part (d) to verify your answers to parts (b) and (c). 65. Physics The function T(t) = T r + (T i - T r )e kt models Newton s Law of Cooling. T(t) is the temperature of a heated substance t minutes after it has been removed from a heat (or cooling) source. T i is the substance s initial temperature, k is a constant for that substance, and T r is room temperature. a. The initial surface temperature of a beef roast is 68F and room temperature is 8F. If k =-0.0, how long will it take for this roast to cool to 008F? b. Write and graph an equation that you can use to check your answer to part (a). Use your graph to complete the table below. a b. See margin. 6. No; using the Change of Base Formula would result in one of the log expressions being written as a quotient of logs, which couldn t then be combined with the other expression to form a single logarithm. lna y 00 b 6d. t, where y 0. is the number of Internet users in millions and t is time in years. e. Substitute the number of users found in (b) and (c) into the equation in (d). Determine whether your answers in years are the same as t for each. Test Prep Multiple Choice Chapter 8 Exponential and Logarithmic Functions 65a. about min T b. t ln ( ) Temperature ( F) Minutes Later Open-Ended Write a real-world problem that you can answer using Newton s Law of Cooling. Then answer it. Check students work. 6. Which expression is equal to ln - 5 ln? C A. ln (-8) B. ln Q 6 5 R C. ln D. ln t (min) O T ( F) 00

20 Extended Response 68. What is the value of x if e x = 85? H F. 5 G. ln 85 H. ln 5? ln J. ln 85 ln ln 69. An investment of $50 will be worth $500 after years of continuous compounding at a fixed interest rate. What is that interest rate? B A..00% B. 5.8% C. 6.9% D. 00% 0. The table shows the values of an investment after the given number of years of continuously compounded interest. Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. 8 Test-Taking Strategies, p. 8 Test-Taking Strategies with Transparencies GO Mixed Review for Help Lesson 8-5 Lesson - Lesson 6- Years Value 0 $ $5.6 $586.6 a. What is the rate of interest? a c. See margin. b. Write an equation to model the growth of the investment. c. To the nearest year, when will the investment be worth $800? Solve each equation.. x = 656. x - = x+ = 0.. log x =. # 5. log 5x + = log 9 - log x + = Find the inverse of each function. Is the inverse a function? 9. See margin.. y = 5x + 8. y = x y =-x The Nut Shop carries 0 different types of nuts. The shop special is the Triple Play, a made-to-order mixture of any three different types of nuts. How many different Triple Plays are possible? 060 possible combinations $65.6 $ [] a. 8% b. A Pe rt 500 e 0.08t c e 0.08t.6 e 0.08t In t ln t 6 N t about 6 years [] correct model, computation error in (b) or (c) [] incorrect model, solved correctly [] correct model, but without work shown in (c) x. y 5 ; yes # x0 8. y ; yes A P int in Time 9. y w "5 x; no The first manned moon landing on July 0, 969, gave scientists a unique opportunity to test their theories about the moon s geologic history. A logarithmic function was used to date lunar rocks. Radioactive rubidium-8 decays into stable strontium-8 at a fixed rate. The ratio r of the two isotopes in a sample can be measured and used in the equation T =-h ln (r ) ln 0.5, where T is the age in years and h is the half-life of rubidium-8,. 0 0 years. For the lunar sample, r was measured at , giving an approximate age of.8 billion years. PHSchool.com For: Information about space exploration Web Code: age-0 Lesson 8-6 Natural Logarithms 5 5