Solving Exponential and Logarithmic Equations

5.5 Solving Exponential and Logarithmic Equations Essential Question How can ou solve exponential and logarithmic equations? Solving Exponential and Logarithmic Equations Work with a partner. Match each equation with the graph of its related sstem of equations. Explain our reasoning. Then use the graph to solve the equation. a. e x = b. ln x = 1 c. x = 3 x d. log x = 1 e. log 5 x = 1 f. x = A. B. C. x x x D. E. F. x x x 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 Exponential and Logarithmic Equations Work with a partner. Look back at the equations in Explorations 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 exponents and logarithms. Communicate Your Answer 3. How can ou solve exponential and logarithmic equations?. Solve each equation using an method. Explain our choice of method. a. 16 x = b. x = x + 1 c. x = 3 x + 1 d. log x = 1 e. ln x = f. log 3 x = 3 Section 5.5 Solving Exponential and Logarithmic Equations 81

5.5 Lesson What You Will Learn Core Vocabular exponential equations, p. 8 logarithmic equations, p. 83 Previous extraneous solution inequalit Solve exponential equations. Solve logarithmic equations. Solve exponential and logarithmic inequalities. Solving Exponential Equations Exponential equations are equations in which variable expressions occur as exponents. The result below is useful for solving certain exponential equations. Core Concept Propert of Equalit for Exponential Equations Algebra If b is a positive real number other than 1, then b x = b if and onl if x =. Example If 3 x = 3 5, then x = 5. If x = 5, then 3 x = 3 5. The preceding propert is useful for solving an exponential 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 exponential equation using the same base, ou can tr to solve the equation b taking a logarithm of each side. Solving Exponential Equations Solve each equation. a. 100 x = ( 10) 1 x 3 b. x = 7 100 1 =? ( 10) 1 1 3 100 =? ( 10) 1 100 = 100 a. 100 x = ( 10) 1 x 3 Write original equation. (10 ) x = (10 1 ) x 3 Rewrite 100 and 1 as powers with base 10. 10 10 x = 10 x + 3 Power of a Power Propert x = x + 3 Propert of Equalit for Exponential Equations x = 1 Solve for x. b. x = 7 Write original equation. log x = log 7 Take log of each side. x = log 7 log b b x = x x.807 Use a calculator. Enter = x 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 x = 7 is about.807. 10 0 Intersection X=.807359 Y=7 3 5 8 Chapter 5 Exponential and Logarithmic Functions

An important application of exponential 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 exponential 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 = 0.06. 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 = 0.06. 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. 0.11 e 0.06t Divide each side b 1. ln 0.11 ln e 0.06t Take natural log of each side. 1.556 0.06t ln e x = log e e x = x 33.8 t Divide each side b 0.06. You should wait about 3 minutes before serving the stew. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. 1. x = 5. 7 9x = 15 3. e 0.3x 7 = 13. WHAT IF? In Example, how long will it take to cool the stew to 100 F when the room temperature is 75 F? Solving Logarithmic Equations Logarithmic equations are equations that involve logarithms of variable expressions. You can use the next propert to solve some tpes of logarithmic equations. Core Concept Propert of Equalit for Logarithmic Equations Algebra If b, x, and are positive real numbers with b 1, then log b x = log b if and onl if x =. Example If log x = log 7, then x = 7. If x = 7, then log x = log 7. The preceding propert implies that if ou are given an equation x =, then ou can exponentiate each side to obtain an equation of the form b x = b. This technique is useful for solving some logarithmic equations. Section 5.5 Solving Exponential and Logarithmic Equations 83

Solving Logarithmic Equations Solve (a) ln(x 7) = ln(x + 5) and (b) log (5x 17) = 3. ln( 7) =? ln( + 5) ln(16 7) =? ln 9 ln 9 = ln 9 a. ln(x 7) = ln(x + 5) Write original equation. x 7 = x + 5 Propert of Equalit for Logarithmic Equations 3x 7 = 5 Subtract x from each side. 3x = 1 Add 7 to each side. x = Divide each side b 3. b. log (5x 17) = 3 Write original equation. log (5 5 17) =? 3 log (5 17) =? 3 log 8 =? 3 Because 3 = 8, log 8 = 3. log (5x 17) = 3 Exponentiate each side using base. 5x 17 = 8 5x = 5 b log b x = x Add 17 to each side. x = 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 extraneous solutions of logarithmic equations. You can do this algebraicall or graphicall. Solve log x + log(x 5) =. Solving a Logarithmic Equation 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 x + log(x 5) = Write original equation. log[x(x 5)] = Product Propert of Logarithms 10 log[x(x 5)] = 10 Exponentiate each side using base 10. x(x 5) = 100 b log b x = x x 10x = 100 Distributive Propert x 10x 100 = 0 Write in standard form. x 5x 50 = 0 Divide each side b. (x 10)(x + 5) = 0 Factor. x = 10 or x = 5 Zero-Product Propert The apparent solution x = 5 is extraneous. So, the onl solution is x = 10. Monitoring Progress Solve the equation. for extraneous solutions. Help in English and Spanish at BigIdeasMath.com 5. ln(7x ) = ln(x + 11) 6. log (x 6) = 5 7. log 5x + log(x 1) = 8. log (x + 1) + log x = 3 8 Chapter 5 Exponential and Logarithmic Functions

STUDY TIP Be sure ou understand that these properties of inequalit are onl true for values of b > 1. Solving Exponential and Logarithmic Inequalities Exponential inequalities are inequalities in which variable expressions occur as exponents, and logarithmic inequalities are inequalities that involve logarithms of variable expressions. To solve exponential and logarithmic inequalities algebraicall, use these properties. Note that the properties are true for and. Exponential Propert of Inequalit: If b is a positive real number greater than 1, then b x > b if and onl if x >, and b x < b if and onl if x <. Logarithmic Propert of Inequalit: If b, x, and are positive real numbers with b > 1, then log b x > log b if and onl if x >, and log b x < log b if and onl if x <. You can also solve an inequalit b taking a logarithm of each side or b exponentiating. Solving an Exponential Inequalit Solve 3 x < 0. 3 x < 0 Write original inequalit. log 3 3 x < log 3 0 x < log 3 0 Take log 3 of each side. log b b x = x The solution is x < log 3 0. Because log 3 0.77, the approximate solution is x <.77. Solve log x. Solving a Logarithmic Inequalit Method 1 Use an algebraic approach. log x Write original inequalit. 10 log 10 x 10 Exponentiate each side using base 10. x 100 b log b x = x Because log x is onl defined when x > 0, the solution is 0 < x 100. Method Use a graphical approach. Graph = log x and = in the same viewing window. Use the intersect feature to determine that the graphs intersect when x = 100. The graph of = log x is on or below the graph of = when 0 < x 100. The solution is 0 < x 100. Monitoring Progress Solve the inequalit. 50 Intersection X=100 Y= 1 Help in English and Spanish at BigIdeasMath.com 9. e x < 10. 10 x 6 > 3 11. log x + 9 < 5 1. ln x 1 > 3 175 Section 5.5 Solving Exponential and Logarithmic Equations 85

5.5 Exercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept 1. COMPLETE THE SENTENCE The equation 3 x 1 = 3 is an example of a(n) equation.. WRITING Compare the methods for solving exponential and logarithmic equations. 3. WRITING When do logarithmic equations have extraneous solutions?. COMPLETE THE SENTENCE If b is a positive real number other than 1, then b x = b if and onl if. Monitoring Progress and Modeling with Mathematics In Exercises 5 16, solve the equation. (See Example 1.) 5. 3x + 5 = 1 x 6. e x = e 3x 1 7. 5 x 3 = 5 x 5 8. 6 x 6 = 36 3x 5 9. 3 x = 7 10. 10 x = 33 11. 100 5x + = ( 10) 1 11 x 1. 51 5x 1 = ( 1 x 8) 13. 5(7) 5x = 60 1. 3() 6x = 99 In Exercises 19 and 0, use Newton s Law of Cooling to solve the problem. (See Example.) 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 = 0.0058. How long do ou have to wait until ou can continue driving? 15. 3e x + 9 = 15 16. e x 7 = 5 17. 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 = 0.067. How long do ou have to wait until ou can carve the turke? In Exercises 1 3, solve the equation. (See Example 3.) 1. ln(x 7) = ln(x + 11). ln(x ) = ln(x + 6) 3. log (3x ) = log 5. log(7x + 3) = log 38 18. 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 0.0003t. After how man ears will onl 5 grams of radium be present? 5. log (x + 8) = 5 6. log 3 (x + 1) = 7. log 7 (x + 9) = 8. log 5 (5x + 10) = 9. log(1x 9) = log 3x 30. log 6 (5x + 9) = log 6 6x 86 Chapter 5 Exponential and Logarithmic Functions 31. log (x x 6) = 3. log 3 (x + 9x + 7) =

In Exercises 33 0, solve the equation. for extraneous solutions. (See Example.) 33. log x + log (x ) = 3 3. log 6 3x + log 6 (x 1) = 3 35. ln x + ln(x + 3) = 36. ln x + ln(x ) = 5 37. log 3 3x + log 3 3 = 38. log ( x) + log (x + 10) = 39. log 3 (x 9) + log 3 (x 3) = 0. log 5 (x + ) + log 5 (x + 1) = ERROR ANALYSIS In Exercises 1 and, describe and correct the error in solving the equation. 1.. log 3 (5x 1) = 3 log 3 (5x 1) = 3 5x 1 = 6 5x = 65 x = 13 log (x + 1) + log x = 3 log [(x + 1)(x)] = 3 log [(x + 1)(x)] = 3 (x + 1)(x) = 6 x + 1x 6 = 0 (x + 16)(x ) = 0 x = 16 or x = 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 Approximate the solution of each equation using the graph. a. 1 5 5 x = 9 b. log 5x = 1 Section 5.5 Solving Exponential and Logarithmic Equations 87 8 = 9 x = 1 5 5 x 1 8 = = log 5x 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 Exercises 7 5, solve the inequalit. (See Examples 5 and 6.) 7. 9 x > 5 8. x 36 9. ln x 3 50. log x < 51. 3 x 5 < 8 5. e 3x + > 11 53. 3 log 5 x + 6 9 5. log 5 x 5 3 55. COMPARING METHODS Solve log 5 x < algebraicall and graphicall. Which method do ou prefer? Explain 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 expressed 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 exceed $10,000? USING TOOLS In Exercises 59 6, use a graphing calculator to solve the equation. 59. ln x = 3 x + 60. log x = 7 x 61. log x = 3 x 3 6. ln x = e x 3 x

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 approximate the solution of the inequalit ln x + 6 > 9. Explain our reasoning. = 9 1 6 = ln x + 6 6 65. OPEN-ENDED Write an exponential equation that has a solution of x =. Then write a logarithmic equation that has a solution of x = 3. 66. THOUGHT PROVOKING Give examples of logarithmic or exponential equations that have one solution, two solutions, and no solutions. x CRITICAL THINKING In Exercises 67 7, solve the equation. 67. x + 3 = 5 3x 1 68. 10 3x 8 = 5 x 69. log 3 (x 6) = log 9 x 70. log x = log 8 x 71. x 1 x + 3 = 0 7. 5 x + 0 5 x 15 = 0 73. WRITING In Exercises 67 70, ou solved exponential and logarithmic equations with different bases. Describe general methods for solving such equations. 7. PROBLEM SOLVING When X-ras of a fixed wavelength strike a material x centimeters thick, the intensit I(x) of the X-ras transmitted through the material is given b I(x) = I 0 e μx, 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. Material Aluminum Copper Lead Value of μ 0.3 3. 3 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 x for which I(x) = 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), explain 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 1 3 77. (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 3.9) 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) 88 Chapter 5 Exponential and Logarithmic Functions