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1 Quiz Tell whether the function represents eponential growth or eponential deca. Eplain our reasoning. (Sections 6.1 and 6.) 1. f () = (.5). = ( 3 8) Simplif the epression. (Sections 6. and 6.3) 5. e 8 e 6. 15e 3 3e 3. = e 0.6. f () = 5e 7. (5e ) 3 8. e ln 9 9. log log 3 81 Rewrite the epression in eponential or logarithmic form. (Section 6.3) 11. log 10 = 5 1. log 1/3 7 = = = Evaluate the logarithm. If necessar, use a calculator and round our answer to three decimal places. (Section 6.3) 15. log ln log 3 Graph the function and its inverse. (Section 6.3) 18. f () = ( 1 9) 19. = ln( 7) 0. f () = log 5 ( + 1) The graph of g is a transformation of the graph of f. Write a rule for g. (Section 6.) 1. f () = log 3. f () = 3 3. f () = log 1/ g g g 6. You purchase an antique lamp for $150. The value of the lamp increases b.15% each ear. Write an eponential model that gives the value (in dollars) of the lamp t ears after ou purchased it. (Section 6.1) 5. A local bank advertises two certificate of deposit (CD) accounts that ou can use to save mone and earn interest. The interest is compounded monthl for both accounts. (Section 6.1) a. You deposit the minimum required amounts in each CD account. How much mone is in each account at the end of its term? How much interest does each account earn? Justif our answers. b. Describe the benefits and drawbacks of each account. 6. The Richter scale is used for measuring the magnitude of an earthquake. The Richter magnitude R is given b R = 0.67 ln E , where E is the energ (in kilowatt-hours) released b the earthquake. Graph the model. What is the Richter magnitude for an earthquake that releases 3,000 kilowatt-hours of energ? (Section 6.) CD Specials.0 % annual interest 36/mo CD $1500 Minimum Balance 3.0 % annual interest 60/mo CD $000 Minimum Balance 36 Chapter 6 Eponential and Logarithmic Functions

2 6.5 Properties of Logarithms Essential Question How can ou use properties of eponents to derive properties of logarithms? Let = log b m and = log b n. The corresponding eponential forms of these two equations are CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to understand and use stated assumptions, definitions, and previousl established results. b = m and b = n. Product Propert of Logarithms Work with a partner. To derive the Product Propert, multipl m and n to obtain mn = b b = b +. The corresponding logarithmic form of mn = b + is log b mn = +. So, log b mn =. Product Propert of Logarithms Quotient Propert of Logarithms Work with a partner. To derive the Quotient Propert, divide m b n to obtain m n = b b = b. The corresponding logarithmic form of m n = b is log m b =. So, n log m b =. Quotient Propert of Logarithms n Power Propert of Logarithms Work with a partner. To derive the Power Propert, substitute b for m in the epression log b m n, as follows. log b m n = log b (b ) n Substitute b for m. = log b b n Power of a Power Propert of Eponents = n Inverse Propert of Logarithms So, substituting log b m for, ou have log b m n =. Power Propert of Logarithms Communicate Your Answer. How can ou use properties of eponents to derive properties of logarithms? 5. Use the properties of logarithms that ou derived in Eplorations 1 3 to evaluate each logarithmic epression. a. log 16 3 b. log c. ln e + ln e 5 d. ln e 6 ln e 5 e. log 5 75 log 5 3 f. log + log 3 Section 6.5 Properties of Logarithms 37

3 6.5 Lesson What You Will Learn Core Vocabular Previous base properties of eponents STUDY TIP These three properties of logarithms correspond to these three properties of eponents. a m a n = a m + n a m a n = am n (a m ) n = a mn Use the properties of logarithms to evaluate logarithms. Use the properties of logarithms to epand or condense logarithmic epressions. Use the change-of-base formula to evaluate logarithms. Properties of Logarithms You know that the logarithmic function with base b is the inverse function of the eponential function with base b. Because of this relationship, it makes sense that logarithms have properties similar to properties of eponents. Core Concept Properties of Logarithms Let b, m, and n be positive real numbers with b 1. Product Propert log b mn = log b m + log b n Quotient Propert log b m n = log b m log b n Power Propert log b m n = n log b m Using Properties of Logarithms Use log and log to evaluate each logarithm. a. log 3 7 b. log 1 c. log 9 COMMON ERROR Note that in general log m b n log b m log b n and log b mn (log b m)(log b n). a. log 3 7 = log 3 log 7 Quotient Propert Use the given values of log 3 and log 7. = 1. Subtract. b. log 1 = log (3 7) Write 1 as 3 7. = log 3 + log 7 Product Propert Use the given values of log 3 and log 7. =.39 Add. c. log 9 = log 7 Write 9 as 7. = log 7 Power Propert (.807) Use the given value log 7. = 5.61 Multipl. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use log and log to evaluate the logarithm. 1. log log log 6 6. log Chapter 6 Eponential and Logarithmic Functions

4 Rewriting Logarithmic Epressions You can use the properties of logarithms to epand and condense logarithmic epressions. Epanding a Logarithmic Epression STUDY TIP When ou are epanding or condensing an epression involving logarithms, ou can assume that an variables are positive. Epand ln 57. ln 57 = ln 57 ln Quotient Propert = ln 5 + ln 7 ln Product Propert = ln ln ln Power Propert Condense log log log 3. Condensing a Logarithmic Epression log log log 3 = log 9 + log 3 log 3 Power Propert = log(9 3 ) log 3 Product Propert = log Quotient Propert = log Simplif. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Epand the logarithmic epression. 5. log ln Condense the logarithmic epression log log 9 8. ln + 3 ln 3 ln 1 Change-of-Base Formula Logarithms with an base other than 10 or e can be written in terms of common or natural logarithms using the change-of-base formula. This allows ou to evaluate an logarithm using a calculator. Core Concept Change-of-Base Formula If a, b, and c are positive real numbers with b 1 and c 1, then log c a = log b a log b c. In particular, log c a = log a log c and log c a = ln a ln c. Section 6.5 Properties of Logarithms 39

5 Changing a Base Using Common Logarithms ANOTHER WAY In Eample, log 3 8 can be evaluated using natural logarithms. log 3 8 = ln 8 ln Notice that ou get the same answer whether ou use natural logarithms or common logarithms in the change-of-base formula. Evaluate log 3 8 using common logarithms. log 3 8 = log 8 log 3 log c a = log a log c Use a calculator. Then divide Evaluate log 6 using natural logarithms. log 6 = ln ln 6 Changing a Base Using Natural Logarithms log c a = ln a ln c Use a calculator. Then divide Solving a Real-Life Problem For a sound with intensit I (in watts per square meter), the loudness L(I ) of the sound (in decibels) is given b the function L(I) = 10 log I I 0 where I 0 is the intensit of a barel audible sound (about 10 1 watts per square meter). An artist in a recording studio turns up the volume of a track so that the intensit of the sound doubles. B how man decibels does the loudness increase? Let I be the original intensit, so that I is the doubled intensit. increase in loudness = L(I ) L(I ) Write an epression. = 10 log I 10 log I I 0 I 0 = 10 ( log I log I I 0 I 0 ) = 10 ( log + log I log I I 0 I 0 ) Substitute. = 10 log Simplif. The loudness increases b 10 log decibels, or about 3 decibels. Monitoring Progress Use the change-of-base formula to evaluate the logarithm. Distributive Propert Product Propert Help in English and Spanish at BigIdeasMath.com 9. log log log log WHAT IF? In Eample 6, the artist turns up the volume so that the intensit of the sound triples. B how man decibels does the loudness increase? 330 Chapter 6 Eponential and Logarithmic Functions

6 6.5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE To condense the epression log 3 + log 3, ou need to use the Propert of Logarithms.. WRITING Describe two was to evaluate log 7 1 using a calculator. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, use log and log to evaluate the logarithm. (See Eample 1.) 3. log 7 3. log 7 8. ln 8 3 = 3 ln 8 + ln 5. log log log log In Eercises 9 1, match the epression with the logarithm that has the same value. Justif our answer. 9. log 3 6 log 3 A. log log 3 6 B. log log 3 C. log log log 3 D. log 3 36 In Eercises 13 0, epand the logarithmic epression. (See Eample.) 13. log 3 1. log log ln ln ln log log 5 3 ERROR ANALYSIS In Eercises 1 and, describe and correct the error in epanding the logarithmic epression. 1. log 5 = (log 5)(log ) In Eercises 3 30, condense the logarithmic epression. (See Eample 3.) 3. log 7 log 10. ln 1 ln 5. 6 ln + ln 6. log + log log log ln ln 9. 5 ln + 7 ln + ln 30. log 3 + log log REASONING Which of the following is not equivalent to log 5? Justif our answer. 3 A B C D log 5 log 5 3 log 5 log log 5 log 5 log 5 3 log 5 log 5 log 5 3 log 5 3. REASONING Which of the following equations is correct? Justif our answer. A log 7 + log 7 = log 7 ( + ) B 9 log log = log 9 C 5 log + 7 log = log D log 9 5 log 9 = log 9 5 Section 6.5 Properties of Logarithms 331

7 In Eercises 33 0, use the change-of-base formula to evaluate the logarithm. (See Eamples and 5.) 33. log 7 3. log log log 8. The intensit of the sound of a certain television advertisement is 10 times greater than the intensit of the television program. B how man decibels does the loudness increase? Intensit of Television Sound 37. log log log log MAKING AN ARGUMENT Your friend claims ou can use the change-of-base formula to graph = log 3 using a graphing calculator. Is our friend correct? Eplain our reasoning.. HOW DO YOU SEE IT? Use the graph to determine the value of log 8 log. = log 6 8 MODELING WITH MATHEMATICS In Eercises 3 and, use the function L(I ) given in Eample The blue whale can produce sound with an intensit that is 1 million times greater than the intensit of the loudest sound a human can make. Find the difference in the decibel levels of the sounds made b a blue whale and a human. (See Eample 6.) During show: Intensit = I During ad: Intensit = 10I 5. REWRITING A FORMULA Under certain conditions, the wind speed s (in knots) at an altitude of h meters above a grass plain can be modeled b the function s(h) = ln 100h. a. B what amount 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) = (log h + ). log e 6. THOUGHT PROVOKING Determine whether the formula log b (M + N) = log b M + log b N is true for all positive, real values of M, N, and b (with b 1). Justif our answer. 7. USING STRUCTURE Use the properties of eponents to prove the change-of-base formula. (Hint: Let = log b a, = log b c, and z = log c a.) 8. CRITICAL THINKING Describe three was to transform the graph of f () = log to obtain the graph of g() = log Justif our answers. Maintaining Mathematical Proficienc Solve the inequalit b graphing. (Section 3.6) Reviewing what ou learned in previous grades and lessons 9. > ( 6) < Solve the equation b graphing the related sstem of equations. (Section 3.5) = ( + 3)( ) = = ( + 3) ( + 7) + 5 = ( + 10) 3 33 Chapter 6 Eponential and Logarithmic Functions

8 6.6 Solving Eponential and Logarithmic Equations Essential Question How can ou solve eponential and logarithmic equations? Solving Eponential and Logarithmic Equations Work with a partner. Match each equation with the graph of its related sstem of equations. Eplain our reasoning. Then use the graph to solve the equation. a. e = b. ln = 1 c. = 3 d. log = 1 e. log 5 = 1 f. = A. B. C. D. E. F. MAKING SENSE OF PROBLEMS To be proficient in math, ou need to plan a solution pathwa rather than simpl jumping into a solution attempt. Solving Eponential and Logarithmic Equations Work with a partner. Look back at the equations in Eplorations 1(a) and 1(b). Suppose ou want a more accurate wa to solve the equations than using a graphical approach. a. Show how ou could use a numerical approach b creating a table. For instance, ou might use a spreadsheet to solve the equations. b. Show how ou could use an analtical approach. For instance, ou might tr solving the equations b using the inverse properties of eponents and logarithms. Communicate Your Answer 3. How can ou solve eponential and logarithmic equations?. Solve each equation using an method. Eplain our choice of method. a. 16 = b. = + 1 c. = d. log = 1 e. ln = f. log 3 = 3 Section 6.6 Solving Eponential and Logarithmic Equations 333

9 6.6 Lesson What You Will Learn Core Vocabular eponential equations, p. 33 logarithmic equations, p. 335 Previous etraneous solution inequalit Solve eponential equations. Solve logarithmic equations. Solve eponential and logarithmic inequalities. Solving Eponential Equations Eponential equations are equations in which variable epressions occur as eponents. The result below is useful for solving certain eponential equations. Core Concept Propert of Equalit for Eponential Equations Algebra If b is a positive real number other than 1, then b = b if and onl if =. Eample If 3 = 3 5, then = 5. If = 5, then 3 = 3 5. The preceding propert is useful for solving an eponential equation when each side of the equation uses the same base (or can be rewritten to use the same base). When it is not convenient to write each side of an eponential equation using the same base, ou can tr to solve the equation b taking a logarithm of each side. Solving Eponential Equations Solve each equation. a. 100 = ( 10) 1 3 b. = 7 Check =? ( 10) =? ( 10) = 100 a. 100 = ( 10) 1 3 Write original equation. (10 ) = (10 1 ) 3 Rewrite 100 and 1 as powers with base = Power of a Power Propert = + 3 Propert of Equalit for Eponential Equations = 1 Solve for. b. = 7 Write original equation. log = log 7 Take log of each side. = log 7 log b b =.807 Use a calculator. Check Enter = and = 7 in a graphing calculator. Use the intersect feature to find the intersection point of the graphs. The graphs intersect at about (.807, 7). So, the solution of = 7 is about Intersection X= Y= Chapter 6 Eponential and Logarithmic Functions

10 An important application of eponential equations is Newton s Law of Cooling. This law states that for a cooling substance with initial temperature T 0, the temperature T after t minutes can be modeled b LOOKING FOR STRUCTURE Notice that Newton's Law of Cooling models the temperature of a cooling bod b adding a constant function, T R, to a decaing eponential function, (T 0 T R )e rt. T = (T 0 T R )e rt + T R where T R is the surrounding temperature and r is the cooling rate of the substance. Solving a Real-Life Problem You are cooking aleecha, an Ethiopian stew. When ou take it off the stove, its temperature is 1 F. The room temperature is 70 F, and the cooling rate of the stew is r = How long will it take to cool the stew to a serving temperature of 100 F? Use Newton s Law of Cooling with T = 100, T 0 = 1, T R = 70, and r = T = (T 0 T R )e rt + T R Newton s Law of Cooling 100 = (1 70)e 0.06t + 70 Substitute for T, T 0, T R, and r. 30 = 1e 0.06t Subtract 70 from each side e 0.06t Divide each side b 1. ln 0.11 ln e 0.06t Take natural log of each side t ln e = log e e = 33.8 t Divide each side b You should wait about 3 minutes before serving the stew. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. 1. = = e = 13. WHAT IF? In Eample, how long will it take to cool the stew to 100ºF when the room temperature is 75ºF? Solving Logarithmic Equations Logarithmic equations are equations that involve logarithms of variable epressions. You can use the net propert to solve some tpes of logarithmic equations. Core Concept Propert of Equalit for Logarithmic Equations Algebra If b,, and are positive real numbers with b 1, then log b = log b if and onl if =. Eample If log = log 7, then = 7. If = 7, then log = log 7. The preceding propert implies that if ou are given an equation =, then ou can eponentiate each side to obtain an equation of the form b = b. This technique is useful for solving some logarithmic equations. Section 6.6 Solving Eponential and Logarithmic Equations 335

11 Solving Logarithmic Equations Solve (a) ln( 7) = ln( + 5) and (b) log (5 17) = 3. Check ln( 7) =? ln( + 5) ln(16 7) =? ln 9 ln 9 = ln 9 a. ln( 7) = ln( + 5) Write original equation. 7 = + 5 Propert of Equalit for Logarithmic Equations 3 7 = 5 Subtract from each side. 3 = 1 Add 7 to each side. = Divide each side b 3. b. log (5 17) = 3 Write original equation. Check log (5 5 17) =? 3 log (5 17) =? 3 log 8 =? 3 Because 3 = 8, log 8 = 3. log (5 17) = 3 Eponentiate each side using base = 8 5 = 5 b log b = Add 17 to each side. = 5 Divide each side b 5. Because the domain of a logarithmic function generall does not include all real numbers, be sure to check for etraneous solutions of logarithmic equations. You can do this algebraicall or graphicall. Solve log + log( 5) =. Solving a Logarithmic Equation Check log( 10) + log(10 5) =? log 0 + log 5 =? log 100 =? = log[ ( 5)] + log( 5 5) =? log( 10) + log( 10) =? Because log( 10) is not defined, 5 is not a solution. log + log( 5) = Write original equation. log[( 5)] = Product Propert of Logarithms 10 log[( 5)] = 10 Eponentiate each side using base 10. ( 5) = 100 b log b = 10 = 100 Distributive Propert = 0 Write in standard form = 0 Divide each side b. ( 10)( + 5) = 0 Factor. = 10 or = 5 Zero-Product Propert The apparent solution = 5 is etraneous. So, the onl solution is = 10. Monitoring Progress Solve the equation. Check for etraneous solutions. Help in English and Spanish at BigIdeasMath.com 5. ln(7 ) = ln( + 11) 6. log ( 6) = 5 7. log 5 + log( 1) = 8. log ( + 1) + log = Chapter 6 Eponential and Logarithmic Functions

12 STUDY TIP Be sure ou understand that these properties of inequalit are onl true for values of b > 1. Solving Eponential and Logarithmic Inequalities Eponential inequalities are inequalities in which variable epressions occur as eponents, and logarithmic inequalities are inequalities that involve logarithms of variable epressions. To solve eponential and logarithmic inequalities algebraicall, use these properties. Note that the properties are true for and. Eponential Propert of Inequalit: If b is a positive real number greater than 1, then b > b if and onl if >, and b < b if and onl if <. Logarithmic Propert of Inequalit: If b,, and are positive real numbers with b > 1, then log b > log b if and onl if >, and log b < log b if and onl if <. You can also solve an inequalit b taking a logarithm of each side or b eponentiating. Solving an Eponential Inequalit Solve 3 < 0. 3 < 0 Write original inequalit. log 3 3 < log 3 0 < log 3 0 Take log 3 of each side. log b b = The solution is < log 3 0. Because log , the approimate solution is <.77. Solve log. Solving a Logarithmic Inequalit Method 1 Use an algebraic approach. log Write original inequalit. 10 log Eponentiate each side using base b log b = Because log is onl defined when > 0, the solution is 0 < 100. Method Use a graphical approach. Graph = log and = in the same viewing window. Use the intersect feature to determine that the graphs intersect when = 100. The graph of = log is on or below the graph of = when 0 < 100. The solution is 0 < 100. Monitoring Progress Solve the inequalit. 50 Intersection X=100 Y= 1 Help in English and Spanish at BigIdeasMath.com 9. e < > log + 9 < 5 1. ln 1 > Section 6.6 Solving Eponential and Logarithmic Equations 337

13 6.6 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The equation 3 1 = 3 is an eample of a(n) equation.. WRITING Compare the methods for solving eponential and logarithmic equations. 3. WRITING When do logarithmic equations have etraneous solutions?. COMPLETE THE SENTENCE If b is a positive real number other than 1, then b = b if and onl if. Monitoring Progress and Modeling with Mathematics In Eercises 5 16, solve the equation. (See Eample 1.) = e = e = = = = = ( 7) = ( 1 8) = = 38 In Eercises 19 and 0, use Newton s Law of Cooling to solve the problem. (See Eample.) 19. You are driving on a hot da when our car overheats and stops running. The car overheats at 80 F and can be driven again at 30 F. When it is 80 F outside, the cooling rate of the car is r = How long do ou have to wait until ou can continue driving? 15. 3e + 9 = e 7 = MODELING WITH MATHEMATICS The length (in centimeters) of a scalloped hammerhead shark can be modeled b the function = 66 19e 0.05t where t is the age (in ears) of the shark. How old is a shark that is 175 centimeters long? 0. You cook a turke until the internal temperature reaches 180 F. The turke is placed on the table until the internal temperature reaches 100 F and it can be carved. When the room temperature is 7 F, the cooling rate of the turke is r = How long do ou have to wait until ou can carve the turke? In Eercises 1 3, solve the equation. (See Eample 3.) 1. ln( 7) = ln( + 11). ln( ) = ln( + 6) 3. log (3 ) = log 5. log(7 + 3) = log MODELING WITH MATHEMATICS One hundred grams of radium are stored in a container. The amount R (in grams) of radium present after t ears can be modeled b R = 100e t. After how man ears will onl 5 grams of radium be present? 5. log ( + 8) = 5 6. log 3 ( + 1) = 7. log 7 ( + 9) = 8. log 5 (5 + 10) = 9. log(1 9) = log log 6 (5 + 9) = log Chapter 6 Eponential and Logarithmic Functions 31. log ( 6) = 3. log 3 ( ) =

14 In Eercises 33 0, solve the equation. Check for etraneous solutions. (See Eample.) 33. log + log ( ) = 3 3. log log 6 ( 1) = ln + ln( + 3) = 36. ln + ln( ) = log log 3 3 = 38. log ( ) + log ( + 10) = 39. log 3 ( 9) + log 3 ( 3) = 0. log 5 ( + ) + log 5 ( + 1) = ERROR ANALYSIS In Eercises 1 and, describe and correct the error in solving the equation. 1.. log 3 (5 1) = 3 log 3 (5 1) = = 6 5 = 65 = 13 log ( + 1) + log = 3 log [( + 1)()] = 3 log [( + 1)()] = 3 ( + 1)() = = 0 ( + 16)( ) = 0 = 16 or = 3. PROBLEM SOLVING You deposit $100 in an account that pas 6% annual interest. How long will it take for the balance to reach $1000 for each frequenc of compounding? a. annual b. quarterl c. dail d. continuousl. MODELING WITH MATHEMATICS The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude M of the dimmest star that can be seen with a telescope is M = 5 log D +, where D is the diameter (in millimeters) of the telescope s objective lens. What is the diameter of the objective lens of a telescope that can reveal stars with a magnitude of 1? 5. ANALYZING RELATIONSHIPS Approimate the solution of each equation using the graph. a = 9 b. log 5 = 1 Section 6.6 Solving Eponential and Logarithmic Equations = 9 = = = log 5 6. MAKING AN ARGUMENT Your friend states that a logarithmic equation cannot have a negative solution because logarithmic functions are not defined for negative numbers. Is our friend correct? Justif our answer. In Eercises 7 5, solve the inequalit. (See Eamples 5 and 6.) 7. 9 > ln log < < 8 5. e 3 + > log log COMPARING METHODS Solve log 5 < algebraicall and graphicall. Which method do ou prefer? Eplain our reasoning. 56. PROBLEM SOLVING You deposit $1000 in an account that pas 3.5% annual interest compounded monthl. When is our balance at least $100? $3500? 57. PROBLEM SOLVING An investment that earns a rate of return r doubles in value in t ears, where ln t = and r is epressed as a decimal. What ln(1 + r) rates of return will double the value of an investment in less than 10 ears? 58. PROBLEM SOLVING Your famil purchases a new car for $0,000. Its value decreases b 15% each ear. During what interval does the car s value eceed $10,000? USING TOOLS In Eercises 59 6, use a graphing calculator to solve the equation. 59. ln = log = log = ln = e 3

15 63. REWRITING A FORMULA A biologist can estimate the age of an African elephant b measuring the length of its footprint and using the equation = 5 5.7e 0.09a, where is the length 36 cm (in centimeters) of the footprint and a is the age (in ears). a. Rewrite the equation, solving for a in terms of. b. Use the equation in part (a) to find the ages of the elephants whose footprints are shown. 3 cm 8 cm cm 6. HOW DO YOU SEE IT? Use the graph to solve the inequalit ln + 6 > 9. Eplain our reasoning. 1 CRITICAL THINKING In Eercises 67 7, solve the equation = = log 3 ( 6) = log log = log = = WRITING In Eercises 67 70, ou solved eponential and logarithmic equations with different bases. Describe general methods for solving such equations. 7. PROBLEM SOLVING When X-ras of a fied wavelength strike a material centimeters thick, the intensit I() of the X-ras transmitted through the material is given b I() = I 0 e μ, where I 0 is the initial intensit and μ is a value that depends on the tpe of material and the wavelength of the X-ras. The table shows the values of μ for various materials and X-ras of medium wavelength. = 9 6 = ln Material Aluminum Copper Lead Value of μ OPEN-ENDED Write an eponential equation that has a solution of =. Then write a logarithmic equation that has a solution of = THOUGHT PROVOKING Give eamples of logarithmic or eponential equations that have one solution, two solutions, and no solutions. a. Find the thickness of aluminum shielding that reduces the intensit of X-ras to 30% of their initial intensit. (Hint: Find the value of for which I() = 0.3I 0.) b. Repeat part (a) for the copper shielding. c. Repeat part (a) for the lead shielding. d. Your dentist puts a lead apron on ou before taking X-ras of our teeth to protect ou from harmful radiation. Based on our results from parts (a) (c), eplain wh lead is a better material to use than aluminum or copper. Maintaining Mathematical Proficienc Write an equation in point-slope form of the line that passes through the given point and has the given slope. (Skills Review Handbook) 75. (1, ); m = 76. (3, ); m = Reviewing what ou learned in previous grades and lessons (3, 8); m = 78. (, 5); m = Use finite differences to determine the degree of the polnomial function that fits the data. Then use technolog to find the polnomial function. (Section.9) 79. ( 3, 50), (, 13), ( 1, 0), (0, 1), (1, ), (, 15), (3, 5), (, 15) 80. ( 3, 139), (, 3), ( 1, 1), (0, ), (1, 1), (, ), (3, 37), (, 16) 81. ( 3, 37), (, 8), ( 1, 17), (0, 6), (1, 3), (, 3), (3, 189), (, 6) 30 Chapter 6 Eponential and Logarithmic Functions

16 6.7 Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match each tpe of model with the appropriate scatter plot. Use a regression program to find a model that fits the scatter plot. a. linear (positive slope) b. linear (negative slope) c. quadratic d. cubic e. eponential f. logarithmic A. B C. D E. F. 6 8 USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. 6 6 Eploring Gaussian and Logistic Models Work with a partner. Two common tpes of functions that are related to eponential functions are given. Use a graphing calculator to graph each function. Then determine the domain, range, intercept, and asmptote(s) of the function. 1 a. Gaussian Function: f () = e b. Logistic Function: f () = 1 + e Communicate Your Answer 3. How can ou recognize polnomial, eponential, and logarithmic models?. Use the Internet or some other reference to find real-life data that can be modeled using one of the tpes given in Eploration 1. Create a table and a scatter plot of the data. Then use a regression program to find a model that fits the data. Section 6.7 Modeling with Eponential and Logarithmic Functions 31

17 6.7 Lesson What You Will Learn Core Vocabular Previous finite differences common ratio point-slope form Classif data sets. Write eponential functions. Use technolog to find eponential and logarithmic models. Classifing Data You have analzed fi nite differences of data with equall-spaced inputs to determine what tpe of polnomial function can be used to model the data. For eponential data with equall-spaced inputs, the outputs are multiplied b a constant factor. So, consecutive outputs form a constant ratio. Classifing Data Sets Determine the tpe of function represented b each table. a b a. The inputs are equall spaced. Look for a pattern in the outputs As increases b 1, is multiplied b. So, the common ratio is, and the data in the table represent an eponential function. REMEMBER First differences of linear functions are constant, second differences of quadratic functions are constant, and so on. b. The inputs are equall spaced. The outputs do not have a common ratio. So, analze the finite differences first differences second differences The second differences are constant. So, the data in the table represent a quadratic function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine the tpe of function represented b the table. Eplain our reasoning Chapter 6 Eponential and Logarithmic Functions

18 Writing Eponential Functions You know that two points determine a line. Similarl, two points determine an eponential curve. Writing an Eponential Function Using Two Points Write an eponential function = ab whose graph passes through (1, 6) and (3, 5). Step 1 Substitute the coordinates of the two given points into = ab. 6 = ab 1 Equation 1: Substitute 6 for and 1 for. 5 = ab 3 Equation : Substitute 5 for and 3 for. REMEMBER You know that b must be positive b the definition of an eponential function. Step Solve for a in Equation 1 to obtain a = 6 and substitute this epression for a b in Equation. 5 = ( 6 b ) b3 Substitute 6 for a in Equation. b 5 = 6b Simplif. 9 = b Divide each side b 6. 3 = b Take the positive square root because b > 0. Step 3 Determine that a = 6 b = 6 3 =. So, the eponential function is = (3 ). Data do not alwas show an eact eponential relationship. When the data in a scatter plot show an approimatel eponential relationship, ou can model the data with an eponential function. Finding an Eponential Model Year, Number of trampolines, A store sells trampolines. The table shows the numbers of trampolines sold during the th ear that the store has been open. Write a function that models the data. Step 1 Make a scatter plot of the data. The data appear eponential. Step Choose an two points to write a model, such as (1, 1) and (, 36). Substitute the coordinates of these two points into = ab. 1 = ab 1 36 = ab Solve for a in the first equation to obtain a = 1 b. Substitute to obtain b = and a = So, an eponential function that models the data is = 8.3(1.). Number of trampolines Trampoline Sales Year Section 6.7 Modeling with Eponential and Logarithmic Functions 33

19 A set of more than two points (, ) fits an eponential pattern if and onl if the set of transformed points (, ln ) fits a linear pattern. Graph of points (, ) Graph of points (, ln ) = 1 (, 3 ( 1 ( 1, 1 ( 1 (0, 1) 1 (1, ) The graph is an eponential curve. ln ln = (ln ) 3 ( 1, 0.69) (0, 0) (1, 0.69) 1 1 (, 1.39) The graph is a line. LOOKING FOR STRUCTURE Because the aes are and ln, the point-slope form is rewritten as ln ln 1 = m( 1 ). The slope of the line through (1,.8) and (7,.56) is Writing a Model Using Transformed Points Use the data from Eample 3. Create a scatter plot of the data pairs (, ln ) to show that an eponential model should be a good fit for the original data pairs (, ). Then write an eponential model for the original data. Step 1 Create a table of data pairs (, ln ) ln Step Plot the transformed points as shown. The points lie close to a line, so an eponential model should be a good fit for the original data. Step 3 Find an eponential model = ab b choosing an two points on the line, such as (1,.8) and (7,.56). Use these points to write an equation of the line. Then solve for. ln.8 = 0.35( 1) Equation of line ln = Simplif. = e Eponentiate each side using base e. = e 0.35 (e.13 ) = 8.1(1.) Use properties of eponents. Simplif. So, an eponential function that models the data is = 8.1(1.). ln 6 8 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an eponential function = ab whose graph passes through the given points. 3. (, 1), (3, ). (1, ), (3, 3) 5. (, 16), (5, ) 6. WHAT IF? Repeat Eamples 3 and using the sales data from another store. Year, Number of trampolines, Chapter 6 Eponential and Logarithmic Functions

20 Using Technolog You can use technolog to find best-fit models for eponential and logarithmic data. Finding an Eponential Model Use a graphing calculator to find an eponential model for the data in Eample 3. Then use this model and the models in Eamples 3 and to predict the number of trampolines sold in the eighth ear. Compare the predictions. Enter the data into a graphing calculator and perform an eponential regression. The model is = 8.6(1.). Substitute = 8 into each model to predict the number of trampolines sold in the eighth ear. Eample 3: = 8.3(1.) 8 15 Eample : = 8.1(1.) Regression model: = 8.6(1.) 8 10 EpReg =a*b^ a= b= r = r= The predictions are close for the regression model and the model in Eample that used transformed points. These predictions are less than the prediction for the model in Eample 3. Finding a Logarithmic Model The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is 1 atmosphere ( kilograms per square centimeter). The table shows the pressures p (in atmospheres) at selected altitudes h (in kilometers). Use a graphing calculator to find a logarithmic model of the form h = a + b ln p that represents the data. Estimate the altitude when the pressure is 0.75 atmosphere. Air pressure, p Altitude, h Weather balloons carr instruments that send back information such as wind speed, temperature, and air pressure. Enter the data into a graphing calculator and perform a logarithmic regression. The model is h = ln p. Substitute p = 0.75 into the model to obtain h = ln So, when the air pressure is 0.75 atmosphere, the altitude is about.7 kilometers. Monitoring Progress LnReg =a+bln a= b= r = r= Help in English and Spanish at BigIdeasMath.com 7. Use a graphing calculator to find an eponential model for the data in Monitoring Progress Question Use a graphing calculator to find a logarithmic model of the form p = a + b ln h for the data in Eample 6. Eplain wh the result is an error message. Section 6.7 Modeling with Eponential and Logarithmic Functions 35

21 6.7 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE Given a set of more than two data pairs (, ), ou can decide whether a(n) function fits the data well b making a scatter plot of the points (, ln ).. WRITING Given a table of values, eplain how ou can determine whether an eponential function is a good model for a set of data pairs (, ). Monitoring Progress and Modeling with Mathematics In Eercises 3 6, determine the tpe of function represented b the table. Eplain our reasoning. (See Eample 1.) In Eercises 7 16, write an eponential function = ab whose graph passes through the given points. (See Eample.) 7. (1, 3), (, 1) 8. (, ), (3, 1) 9. (3, 1), (5, ) 10. (3, 7), (5, 3) 11. (1, ), (3, 50) 1. (1, 0), (3, 60) 13. ( 1, 10), (, 0.31) 1. (, 6.), (5, 09.6) (, ) (1, 0.5) ( 3, 10.8) 6 (, 3.6) 1 8 ERROR ANALYSIS In Eercises 17 and 18, describe and correct the error in determining the tpe of function represented b the data The outputs have a common ratio of 3, so the data represent a linear function The outputs have a common ratio of, so the data represent an eponential function. 19. MODELING WITH MATHEMATICS A store sells motorized scooters. The table shows the numbers of scooters sold during the th ear that the store has been open. Write a function that models the data. (See Eample 3.) Chapter 6 Eponential and Logarithmic Functions

22 0. MODELING WITH MATHEMATICS The table shows the numbers of visits to a website during the th month. Write a function that models the data. Then use our model to predict the number of visits after 1 ear In Eercises 1, determine whether the data show an eponential relationship. Then write a function that models the data MODELING WITH MATHEMATICS Your visual near point is the closest point at which our ees can see an object distinctl. The diagram shows the near point (in centimeters) at age (in ears). Create a scatter plot of the data pairs (, ln ) to show that an eponential model should be a good fit for the original data pairs (, ). Then write an eponential model for the original data. (See Eample.) Visual Near Point Distances Age 0 1 cm Age cm Age 0 5 cm Age 50 0 cm Age cm 6. MODELING WITH MATHEMATICS Use the data from Eercise 19. Create a scatter plot of the data pairs (, ln ) to show that an eponential model should be a good fit for the original data pairs (, ). Then write an eponential model for the original data. In Eercises 7 30, create a scatter plot of the points (, ln ) to determine whether an eponential model fits the data. If so, find an eponential model for the data USING TOOLS Use a graphing calculator to find an eponential model for the data in Eercise 19. Then use the model to predict the number of motorized scooters sold in the tenth ear. (See Eample 5.) 3. USING TOOLS A doctor measures an astronaut s pulse rate (in beats per minute) at various times (in minutes) after the astronaut has finished eercising. The results are shown in the table. Use a graphing calculator to find an eponential model for the data. Then use the model to predict the astronaut s pulse rate after 16 minutes Section 6.7 Modeling with Eponential and Logarithmic Functions 37

23 33. USING TOOLS An object at a temperature of 160 C is removed from a furnace and placed in a room at 0 C. The table shows the temperatures d (in degrees Celsius) at selected times t (in hours) after the object was removed from the furnace. Use a graphing calculator to find a logarithmic model of the form t = a + b ln d that represents the data. Estimate how long it takes for the object to cool to 50 C. (See Eample 6.) d t HOW DO YOU SEE IT? The graph shows a set of data points (, ln ). Do the data pairs (, ) fit an eponential pattern? Eplain our reasoning. (, 1) (, 3) ln (0, 1) (, 3) 3. USING TOOLS The f-stops on a 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. The table shows several f-stops on a 35-millimeter camera. Use a graphing calculator to find a logarithmic model of the form s = a + b ln f that represents the data. Estimate the amount of light that strikes the film when f = f s MAKING AN ARGUMENT Your friend sas it is possible to find a logarithmic model of the form d = a + b ln t for the data in Eercise 33. Is our friend correct? Eplain. 38. THOUGHT PROVOKING Is it possible to write as an eponential function of? Eplain our reasoning. (Assume p is positive.) 1 p p 3 p 8p 5 16p 35. DRAWING CONCLUSIONS The table shows the average weight (in kilograms) of an Atlantic cod that is ears old from the Gulf of Maine. Age, Weight, a. Show that an eponential model fits the data. Then find an eponential model for the data. b. B what percent does the weight of an Atlantic cod increase each ear in this period of time? Eplain. Maintaining Mathematical Proficienc Tell whether and are in a proportional relationship. Eplain our reasoning. (Skills Review Handbook) 0. = 1. = 3 1. = CRITICAL THINKING You plant a sunflower seedling in our garden. The height h (in centimeters) of the seedling after t weeks can be modeled b the logistic function 56 h(t) = e 0.65t. a. Find the time it takes the sunflower seedling to reach a height of 00 centimeters. b. Use a graphing calculator to graph the function. Interpret the meaning of the asmptote in the contet of this situation. Reviewing what ou learned in previous grades and lessons 3. = Identif the focus, directri, and ais of smmetr of the parabola. Then graph the equation. (Section.3). = = 6. = 3 7. = 5 38 Chapter 6 Eponential and Logarithmic Functions

24 What Did You Learn? Core Vocabular eponential equations, p. 33 logarithmic equations, p. 335 Core Concepts Section 6.5 Properties of Logarithms, p. 38 Change-of-Base Formula, p. 39 Section 6.6 Propert of Equalit for Eponential Equations, p. 33 Propert of Equalit for Logarithmic Equations, p. 335 Section 6.7 Classifing Data, p. 3 Writing Eponential Functions, p. 33 Using Eponential and Logarithmic Regression, p. 35 Mathematical Practices 1. Eplain how ou used properties of logarithms to rewrite the function in part (b) of Eercise 5 on page 33.. How can ou use cases to analze the argument given in Eercise 6 on page 339? Performance Task Measuring Natural Disasters In 005, an earthquake measuring.1 on the Richter scale barel shook the cit of Ocotillo, California, leaving virtuall no damage. But in 1906, an earthquake with an estimated 8. on the same scale devastated the cit of San Francisco. Does twice the measurement on the Richter scale mean twice the intensit of the earthquake? To eplore the answers to these questions and more, go to BigIdeasMath.com. 39

25 ( 6 Chapter Review 6.1 Eponential Growth and Deca Functions (pp ) Dnamic Solutions available at BigIdeasMath.com Tell whether the function = 3 represents eponential growth or eponential deca. Then graph the function. Step 1 Identif the value of the base. The base, 3, is greater than 1, so the function represents eponential growth. Step Make a table of values Step 3 Step Plot the points from the table. Draw, from left to right, a smooth curve that begins just above the -ais, passes through the plotted points, and moves up to the right. Tell whether the function represents eponential growth or eponential deca. Identif the percent increase or decrease. Then graph the function. 1 ( 1, 3 1 (, 9 ( (, 9) 8 6 (1, 3) (0, 1) 1. f () = ( 1 3 ). = 5 3. f () = (0.). You deposit $1500 in an account that pas 7% annual interest. Find the balance after ears when the interest is compounded dail. 6. The Natural Base e (pp ) Simplif each epression. a. 18e13 e 7 = 9e13 7 = 9e 6 b. (e 3 ) 3 = 3 (e 3 ) 3 = 8e 9 Simplif the epression. 5. e e e 3 10e 6 7. ( 3e 5 ) Tell whether the function represents eponential growth or eponential deca. Then graph the function. 8. f () = 1 3 e 9. = 6e 10. = 3e Logarithms and Logarithmic Functions (pp ) Find the inverse of the function = ln( ). = ln( ) Write original function. = ln( ) Switch and. e = Write in eponential form. e + = Add to each side. Check The graphs appear to be reflections of each other in the line =. The inverse of = ln( ) is = e Chapter 6 Eponential and Logarithmic Functions

26 Evaluate the logarithm. 11. log 8 1. log log 5 1 Find the inverse of the function. 1. f () = = ln( ) 16. = log( + 9) 17. Graph = log 1/5. 6. Transformations of Eponential and Logarithmic Functions (pp ) Describe the transformation of f () = ( 1 g() = ( 1 3) 1 3) represented b + 3. Then graph each function. Notice that the function is of the form g() = ( 1 3) h + k, where h = 1 and k = 3. f g So, the graph of g is a translation 1 unit right and 3 units up of the graph of f. Describe the transformation of f represented b g. Then graph each function. 18. f () = e, g() = e f () = log, g() = 1 log ( + 5) Write a rule for g. 0. Let the graph of g be a vertical stretch b a factor of 3, followed b a translation 6 units left and 3 units up of the graph of f () = e. 1. Let the graph of g be a translation units down, followed b a reflection in the -ais of the graph of f () = log. 6.5 Properties of Logarithms (pp ) Epand ln 15. ln 15 = ln 1 5 ln Quotient Propert = ln 1 + ln 5 ln Product Propert = ln ln ln Power Propert Epand or condense the logarithmic epression.. log log ln log 7 + log log 1 log 7. ln + 5 ln ln 8 Use the change-of-base formula to evaluate the logarithm. 8. log log log 3 Chapter 6 Chapter Review 351

27 6.6 Solving Eponential and Logarithmic Equations (pp ) Solve ln(3 9) = ln( + 6). ln(3 9) = ln( + 6) 3 9 = = 6 = 15 Write original equation. Propert of Equalit for Logarithmic Equations Subtract from each side. Add 9 to each side. Check ln(3 15 9) =? ln( ) ln(5 9) =? ln(30 + 6) ln 36 = ln 36 Solve the equation. Check for etraneous solutions = 8 3. log 3 ( 5) = 33. ln + ln( + ) = 3 Solve the inequalit > ln e Modeling with Eponential and Logarithmic Functions (pp ) Write an eponential function whose graph passes through (1, 3) and (, ). Step 1 Substitute the coordinates of the two given points into = ab. 3 = ab 1 Equation 1: Substitute 3 for and 1 for. = ab Equation : Substitute for and for. Step Solve for a in Equation 1 to obtain a = 3 and substitute this epression for a in Equation. b = ( 3 b ) b Substitute 3 for a in Equation. b = 3b 3 Simplif. 8 = b 3 Divide each side b 3. = b Take cube root of each side. Step 3 Determine that a = 3 b = 3. So, the eponential function is = 3 ( ). Write an eponential model for the data pairs (, ). 37. (3, 8), (5, ) ln A shoe store sells a new tpe of basketball shoe. The table shows the pairs sold s over time t (in weeks). Use a graphing calculator to find a logarithmic model of the form s = a + b ln t that represents the data. Estimate how man pairs of shoes are sold after 6 weeks. Week, t Pairs sold, s Chapter 6 Eponential and Logarithmic Functions

28 6 Chapter Test Graph the equation. State the domain, range, and asmptote. ( 1 ) 1. = 3. = e. = log1/5 Describe the transformation of f represented b g. Then write a rule for g.. f () = log ( 1 ) 5. f () = e 6. f () = 5 g g 1 g Evaluate the logarithm. Use log3 1.6 and log , if necessar log log log3 8 + log3 9. log Describe the similarities and differences in solving the equations 5 = 16 and log(10 + 6) = 1. Then solve each equation. log 11 log 5 ln 11 ln 5 1. Without calculating, determine whether log511,, and are equivalent epressions. Eplain our reasoning. 13. The amount of oil collected b a petroleum compan drilling on the U.S. continental shelf can be modeled b = 1.63 ln 5.381, where is measured in billions of barrels and is the number of wells drilled. About how man barrels of oil would ou epect to collect after drilling 1000 wells? Find the inverse function and describe the information ou obtain from finding the inverse. 1. The percent L of surface light that filters down through bodies of water can be modeled b the eponential function L() = 100e k, where k is a measure of the murkiness of the water and is the depth (in meters) below the surface. a. A recreational submersible is traveling in clear water with a k-value of about 0.0. Write a function that gives the percent of surface light that filters down through clear water as a function of depth. b. Tell whether our function in part (a) represents eponential growth or eponential deca. Eplain our reasoning. c. Estimate the percent of surface light available at a depth of 0 meters. 0m 10 m L = 8% 0 m L = 67% 30 m L = 55% 0 m 15. The table shows the values (in dollars) of a new snowmobile after ears of ownership. Describe three different was to find an eponential model that represents the data. Then write and use a model to find the ear when the snowmobile is worth $500. Year, Value, Chapter 6 hsnb_alg_pe_06ec.indd 353 Chapter Test 353 /5/15 11:31 AM

29 6 Cumulative Assessment 1. Select ever value of b for the equation = b that could result in the graph shown. = b e.0 e 1/ 5. Your friend claims more interest is earned when an account pas interest compounded continuousl than when it pas interest compounded dail. Do ou agree with our friend? Justif our answer. 3. You are designing a rectangular picnic cooler with a length four times its width and height twice its width. The cooler has insulation that is 1 inch thick on each of the four sides and inches thick on the top and bottom. 1 in. in. a. Let represent the width of the cooler. Write a polnomial function T that gives the volume of the rectangular prism formed b the outer surfaces of the cooler. b. Write a polnomial function C for the volume of the inside of the cooler. c. Let I be a polnomial function that represents the volume of the insulation. How is I related to T and C? d. Write I in standard form. What is the volume of the insulation when the width of the cooler is 8 inches?. What is the solution to the logarithmic inequalit log 0? A 3 B 0 3 C 0 < 3 D 3 35 Chapter 6 Eponential and Logarithmic Functions