Solve Exponential and Logarithmic Equations. You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations.

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1 TEKS 7.6 Solve Exponential and Logarithmic Equations 2A..A, 2A..C, 2A..D, 2A..F Before Now You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations. Why? So you can solve problems about astronomy, as in Example 7. Key Vocabulary exponential equation logarithmic equation extraneous solution, p. 52 Exponential equations are equations in which variable expressions occur as exponents. The result below is useful for solving certain exponential equations. KEY CONCEPT Property of Equality for Exponential Equations Algebra For Your Notebook If b is a positive number other than, then b x 5 b y if and only if x 5 y. Example If 3 x 5 3 5, then x 5 5. If x 5 5, then 3 x E XAMPLE Solve by equating exponents Solve 4 x 5 } 2 2 x x 5 } 2 2 x 2 3 Write original equation. (2 2 ) x 5 (2 2 ) x 2 3 Rewrite 4 and } 2 as powers with base x 5 2 2x 3 Power of a power property 2x 52x 3 Property of equality for exponential equations x 5 Solve for x. c The solution is. CHECK Check the solution by substituting it into the original equation. 4 0 } Substitute for x. 40 } Simplify Solution checks. GUIDED PRACTICE for Example Solve the equation.. 9 2x 5 27 x x x x 5 } 3 2 5x Solve Exponential and Logarithmic Equations 55

2 When it is not convenient to write each side of an exponential equation using the same base, you can solve the equation by taking a logarithm of each side. E XAMPLE 2 Take a logarithm of each side ANOTHER WAY For an alternative method for solving the problem in Example 2, turn to page 523 for the Problem Solving Workshop. Solve 4 x 5. 4 x 5 Write original equation. log 4 4 x 5 log 4 x 5 log 4 x 5 Take log 4 of each side. log b b x 5 x log } log 4 Change-of-base formula x ø.73 Use a calculator. c The solution is about.73. Check this in the original equation. NEWTON S LAW OF COOLING 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 by T 5 (T 0 2 T R )e 2rt T R where T R is the surrounding temperature and r is the substance s cooling rate. E XAMPLE 3 Use an exponential model CARS You are driving on a hot day when your car overheats and stops running. It overheats at 2808F and can be driven again at 2308F. If r and it is 808F outside, how long (in minutes) do you have to wait until you can continue driving? Solution T 5 (T 0 2 T R )e 2rt T R Newton s law of cooling ( )e t 80 Substitute for T, T 0, T R, and r e t Subtract 80 from each side e t Divide each side by 200. ln ln e t Take natural log of each side ø t ln e x 5 log e e x 5 x 60 ø t Divide each side by c You have to wait about 60 minutes until you can continue driving. GUIDED PRACTICE for Examples 2 and 3 Solve the equation x x e 20.3x Chapter 7 Exponential and Logarithmic Functions

3 SOLVING LOGARITHMIC EQUATIONS Logarithmic equations are equations that involve logarithms of variable expressions. You can use the following property to solve some types of logarithmic equations. KEY CONCEPT For Your Notebook Property of Equality for Logarithmic Equations Algebra If b, x, and y are positive numbers with b Þ, then log b x 5 log b y if and only if x 5 y. Example If log 2 x 5 log 2 7, then x 5 7. If x 5 7, then log 2 x 5 log 2 7. E XAMPLE 4 Solve a logarithmic equation Solve log 5 (4x 2 7) 5 log 5 (x 5). log 5 (4x 2 7) 5 log 5 (x 5) 4x x 5 3x x 5 2 Write original equation. Property of equality for logarithmic equations Subtract x from each side. Add 7 to each side. x 5 4 Divide each side by 3. c The solution is 4. CHECK Check the solution by substituting it into the original equation. log 5 (4x 2 7) 5 log 5 (x 5) Write original equation. log 5 (4 p 4 2 7) 0 log 5 (4 5) Substitute 4 for x. log log 5 9 Solution checks. EXPONENTIATING TO SOLVE EQUATIONS The property of equality for exponential equations on page 55 implies that if you are given an equation x 5 y, then you can exponentiate each side to obtain an equation of the form b x 5 b y. This technique is useful for solving some logarithmic equations. E XAMPLE 5 Exponentiate each side of an equation Solve log 4 (5x 2 ) 5 3. log 4 (5x 2 ) 5 3 Write original equation. 4 log 4 (5x 2 ) Exponentiate each side using base 4. 5x x 5 65 b log b x 5 x Add to each side. x 5 3 Divide each side by 5. c The solution is 3. CHECK log 4 (5x 2 ) 5 log 4 (5 p 3 2 ) 5 log 4 64 Because , log Solve Exponential and Logarithmic Equations 57

4 EXTRANEOUS SOLUTIONS Because the domain of a logarithmic function generally does not include all real numbers, be sure to check for extraneous solutions of logarithmic equations. You can do this algebraically or graphically. E XAMPLE 6 TAKS PRACTICE: Multiple Choice ELIMINATE CHOICES Instead of solving the equation in Example 6 directly, you can substitute each possible answer into the equation to see whether it is a solution. What is (are) the solution(s) of log 8x log (x 2 20) 5 3? A 25, 25 B 5 C 25 D 5, 25 Solution log 8x log (x 2 20) 5 3 Write original equation. log [8x(x 2 20)] 5 3 Product property of logarithms 0 log [8x(x 2 20)] Exponentiate each side using base 0. 8x(x 2 20) x x x x b log b x 5 x Distributive property Write in standard form. x x Divide each side by 8. (x 2 25)(x 5) 5 0 Factor. x 5 25 or x 525 Zero product property CHECK Check the apparent solutions 25 and 25 using algebra or a graph. Algebra Substitute 25 and 25 for x in the original equation. log 8x log (x 2 20) 5 3 log 8x log (x 2 20) 5 3 log (8 p 25) log ( ) 0 3 log [8(25)] log ( ) 0 3 log 200 log log (240) log (225) 0 3 log Because log (240) and log (225) are not defined, 25 is not a solution. So, 25 is a solution. Graph Graph y 5 log 8x log (x 2 20) and y 5 3 in the same coordinate plane. The graphs intersect only once, when x So, 25 is the only solution. c The correct answer is C. A B C D Intersection X=25 Y=3 GUIDED PRACTICE for Examples 4, 5, and 6 Solve the equation. Check for extraneous solutions. 7. ln (7x 2 4) 5 ln (2x ) 8. log 2 (x 2 6) log 5x log (x 2 ) log 4 (x 2) log 4 x Chapter 7 Exponential and Logarithmic Functions

5 E XAMPLE 7 Use a logarithmic model ASTRONOMY 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 given by the function M 5 5 log D 2 where D is the diameter (in millimeters) of the telescope s objective lens. If a telescope can reveal stars with a magnitude of 2, what is the diameter of its objective lens? ANOTHER WAY For an alternative method for solving the problem in Example 7, turn to page 523 for the Problem Solving Workshop. Solution M 5 5 log D 2 Write original equation log D 2 Substitute 2 for M log D Subtract 2 from each side. 25 log D Divide each side by log D Exponentiate each side using base D Simplify. c The diameter is 00 millimeters. at classzone.com GUIDED PRACTICE for Example 7. WHAT IF? Use the information from Example 7 to find the diameter of the objective lens of a telescope that can reveal stars with a magnitude of EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Exs. 5, 35, and 57 5 TAKS PRACTICE AND REASONING Exs. 44, 47, 58, 60, 62, and 63 5 MULTIPLE REPRESENTATIONS Ex. 59. VOCABULARY Copy and complete: The equation 5 x 5 8 is an example of a(n)? equation. 2. WRITING When do logarithmic equations have extraneous solutions? EXAMPLE on p. 55 for Exs. 3 SOLVING EXPONENTIAL EQUATIONS Solve the equation x x x x 5. 8 x x x x x x x x x 2 5 } x x } x x 8 5 } x 7.6 Solve Exponential and Logarithmic Equations 59

6 EXAMPLE 2 on p. 56 for Exs SOLVING EXPONENTIAL EQUATIONS Solve the equation x e 2x x x x e 22x x e 2x x } 3 (6) 24x x } 4 e 2x 7 } EXAMPLE 4 on p. 57 for Exs EXAMPLES 5 and 6 on pp for Exs SOLVING LOGARITHMIC EQUATIONS Solve the equation. Check for extraneous solutions. 24. log 5 (5x 9) 5 log 5 6x 25. ln (4x 2 7) 5 ln (x ) 26. ln (x 9) 5 ln (7x 2 8) 27. log 5 (2x 2 7) 5 log 5 (3x 2 9) 28. log (2x 2 ) 5 log (3x 3) 29. log 3 (8x 7) 5 log 3 (3x 38) 30. log 6 (3x 2 0) 5 log 6 (4 2 5x) 3. log 8 (5 2 2x) 5 log 8 (6x 2 ) EXPONENTIATING TO SOLVE EQUATIONS Solve the equation. Check for extraneous solutions. 32. log 4 x ln x } 3 log 5 2x log 4 2x log 2 (x 2 4) log 2 x log 2 (x 2 2) log 4 (2x) log 4 (x 0) ln (x 3) ln x ln (2x) log 5 (x 4) log 5 (x ) log 6 3x log 6 (x 2 ) log 3 (x 2 9) log 3 (x 2 3) MULTIPLE TAKS REASONING CHOICE What is the solution of 3 log 8 (2x 7) 8 5 0? A 2.5 B 2.79 C 4 D ERROR ANALYSIS Describe and correct the error in solving the equation x 5 6 x 46. log 3 0x 5 5 log 3 3 x 5 log 3 6 x e log 3 0x 5 e 5 x 5 x log 3 6 0x 5 e 5 x 5 2x 5 x x5 e5 } OPEN-ENDED TAKS REASONING MATH Give an example of an exponential equation whose only solution is 4 and an example of a logarithmic equation whose only solution is 23. CHALLENGE Solve the equation x x x x 50. log 2 (x ) 5 log 8 3x 5. log 3 x 5 log 9 6x x 2 2 p 2 x x 20 p 5 x WORKED-OUT SOLUTIONS 520 Chapter 7 Exponential p. WS and Logarithmic Functions 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

7 PROBLEM SOLVING EXAMPLE 3 on p. 56 for Exs COOKING You are cooking beef stew. When you take the beef stew off the stove, it has a temperature of 2008F. The room temperature is 758F and the cooling rate of the beef stew is r How long (in minutes) will it take to cool the beef stew to a serving temperature of 008F? 55. THERMOMETER As you are hanging an outdoor thermometer, its reading drops from the indoor temperature of 758F to 378F in one minute. If the cooling rate is r 5.37, what is the outdoor temperature? 56. COMPOUND INTEREST You deposit $00 in an account that pays 6% annual interest. How long will it take for the balance to reach $000 for each given frequency of compounding? a. Annual b. Quarterly c. Daily 57. RADIOACTIVE DECAY One hundred grams of radium are stored in a container. The amount R (in grams) of radium present after t years can be modeled by R 5 00e t. After how many years will only 5 grams of radium be present? 58. MULTIPLE TAKS REASONING CHOICE You deposit $800 in an account that pays 2.25% annual interest compounded continuously. About how long will it take for the balance to triple? A 24 years C 48.8 years B 36 years D 52.6 years EXAMPLE 7 on p. 59 for Ex MULTIPLE REPRESENTATIONS The Richter scale is used for measuring the magnitude of an earthquake. The Richter magnitude R is given by the function R log (0.37E).46 where E is the energy (in kilowatt-hours) released by the earthquake. USA GREECE JAPAN Ocotillo Wells, CA May 20, 2005 R = 4. Athens Sept. 7, 999 R = 5.9 Fukuoka March 20, 2005 R = 6.6 a. Making a Graph Graph the function using a graphing calculator. Use your graph to approximate the amount of energy released by each earthquake indicated in the diagram above. b. Solving Equations Write and solve a logarithmic equation to find the amount of energy released by each earthquake in the diagram. 7.6 Solve Exponential and Logarithmic Equations 52

8 60. EXTENDED TAKS REASONING RESPONSE If X-rays of a fixed wavelength strike a material x centimeters thick, then the intensity I(x) of the X-rays transmitted through the material is given by I(x) 5 I 0 e 2μx, where I 0 is the initial intensity and μ is a number that depends on the type of material and the wavelength of the X-rays. The table shows the values of μ for various materials. These μ-values apply to X-rays of medium wavelength. Material Aluminum Copper Lead Value of μ a. Find the thickness of aluminum shielding that reduces the intensity of X-rays to 30% of their initial intensity. (Hint: Find the value of x for which I(x) 5 0.3I 0.) b. Repeat part (a) for copper shielding. c. Repeat part (a) for lead shielding. d. Reasoning Your dentist puts a lead apron on you before taking X-rays of your teeth to protect you from harmful radiation. Based on your results from parts (a) (c), explain why lead is a better material to use than aluminum or copper. 6. CHALLENGE You plant a sunflower seedling in your garden. The seedling s height h (in centimeters) after t weeks can be modeled by the function below, which is called a logistic function. h(t) } 3e 20.65t Find the time it takes the sunflower seedling to reach a height of 200 centimeters. Height (cm) h t Weeks MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 4.; TAKS Workbook 62. TAKS PRACTICE Which list shows the functions in order from the widest graph to the narrowest graph? TAKS Obj. 5 A y 525x 2, y 52 2 } 3 x 2, y 5 5 } 6 x 2, y 5 8x 2 B y 52 2 } 3 x 2, y 5 5 } 6 x 2, y 525x 2, y 5 8x 2 C y 5 5 } 6 x 2, y 52 2 } 3 x 2, y 5 8x 2, y 525x 2 D y 5 8x 2, y 5 5 } 6 x 2, y 52 2 } 3 x 2, y 525x 2 REVIEW Skills Review Handbook p. 994; TAKS Workbook 63. TAKS PRACTICE In the diagram, m 2 5 m 3. What is m? TAKS Obj. 6 F 368 G 648 H 748 J Chapter 7 EXTRA Exponential PRACTICE and Logarithmic for Lesson Functions 7.6, p. 06 ONLINE QUIZ at classzone.com

9 LESSON 7.6 TEKS a.5, a.6, 2A..D, 2A..F Using ALTERNATIVE METHODS Another Way to Solve Examples 2 and 7, pp. 56 and 59 MULTIPLE REPRESENTATIONS In Examples 2 and 7 on pages 56 and 59, respectively, you solved exponential and logarithmic equations algebraically. You can also solve such equations using tables and graphs. P ROBLEM Solve the following exponential equation: 4 x 5. M ETHOD Using a Table One way to solve the equation is to make a table of values. STEP Enter the function y 5 4 x into a graphing calculator. STEP 2 Create a table of values for the function. Y=4^X Y2= Y3= Y4= Y5= Y6= Y7= X Y X=.7 STEP 3 Scroll through the table to find when y 5. The table in Step 2 shows that y 5 between x 5.7 and x 5.8. c The solution of 4 x 5 is between.7 and.8. M ETHOD 2 Using a Graph You can also use a graph to solve the equation. STEP Enter the functions y 5 4 x and y 5 into a graphing calculator. Y=4^X Y2= Y3= Y4= Y5= Y6= Y7= STEP 2 Graph the functions. Use the intersect feature to find the intersection point of the graphs. The graphs intersect at about (.73, ). Use a viewing window of 0 x 5 and 0 y 20. Intersection X= Y= c The solution of 4 x 5 is about.73. Using Alternative Methods 523

10 P ROBLEM 2 ASTRONOMY 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 given by the function M 5 5 log D 2 where D is the diameter (in millimeters) of the telescope s objective lens. If a telescope can reveal stars with a magnitude of 2, what is the diameter of its objective lens? M ETHOD Using a Table Notice that the problem requires solving the following logarithmic equation: 5 log D One way to solve this equation is to make a table of values. You can use a graphing calculator to make the table. STEP Enter the function y 5 5 log x 2 into a graphing calculator. Y=5*log(X)+2 Y2= Y3= Y4= Y5= Y6= Y7= STEP 2 Create a table of values for the function. Make sure that the x-values are in the domain of the function (x > 0). X X= Y STEP 3 Scroll through the table of values to find when y 5 2. The table shows that y 5 2 when x X X=00 Y c To reveal stars with a magnitude of 2, a telescope must have an objective lens with a diameter of 00 millimeters. 524 Chapter 7 Exponential and Logarithmic Functions

11 M ETHOD 2 Using a Graph You can also use a graph to solve the equation 5 log D STEP Enter the functions y 5 5 log x 2 and y 5 2 into a graphing calculator. Y=5*log(X)+2 Y2=2 Y3= Y4= Y5= Y6= Y7= STEP 2 Graph the functions. Use the intersect feature to find the intersection point of the graphs. The graphs intersect at (00, 2). Use a viewing window of 0 x 50 and 0 y 20. Intersection X=00 Y=2 c To reveal stars with a magnitude of 2, a telescope must have an objective lens with a diameter of 00 millimeters. P RACTICE EXPONENTIAL EQUATIONS Solve the equation using a table and using a graph e 3x x e 5x (3) 24x LOGARITHMIC EQUATIONS Solve the equation using a table and using a graph. 5. log 2 5x log (23x 7) ln x log (x 9) ECONOMICS From 998 to 2003, the United States gross national product y (in billions of dollars) can be modeled by y (.04) x where x is the number of years since 998. Use a table and a graph to find the year when the gross national product was $0 trillion. 0. WRITING In Method of Problem on page 523, explain how you could use a table to find the solution of 4 x 5 more precisely.. WHAT IF? In Problem 2 on page 524, suppose the telescope can reveal stars of magnitude 4. Find the diameter of the telescope s objective lens using a table and using a graph. 2. FINANCE You deposit $5000 in an account that pays 3% annual interest compounded quarterly. How long will it take for the balance to reach $6000? Solve the problem using a table and using a graph. 3. OCEANOGRAPHY The density d (in grams per cubic centimeter) of seawater with a salinity of 30 parts per thousand is related to the water temperature T (in degrees Celsius) by the following equation: 0.226T d e For deep water in the South Atlantic Ocean off Antarctica, d g/cm 3. Use a table and a graph to find the water s temperature. Using Alternative Methods 525

12 Extension Use after Lesson 7.6 Solve Exponential and Logarithmic Inequalities TEKS 2A..E, 2A..F GOAL Solve exponential and logarithmic inequalities using tables and graphs. In the Problem Solving Workshop on pages , you learned how to solve exponential and logarithmic equations using tables and graphs. You can use these same methods to solve exponential and logarithmic inequalities. E XAMPLE Solve an exponential inequality CARS Your family purchases a new car for $20,000. Its value decreases by 5% each year. During what interval of time does the car s value exceed $0,000? Solution Let y represent the value of the car (in dollars) x years after it is purchased. A function relating x and y is y 5 20,000( 2 0.5) x, or y 5 20,000(0.85) x. To find the values of x for which y > 0,000, solve the inequality 20,000(0.85) x > 0,000. METHOD Use a table STEP Enter the function y 5 20,000(0.85) x into a graphing calculator. Set the starting x-value of the table to 0 and the step value to 0.. TABLE SETUP TblStart=0 ntbl=0. Indpnt: Auto Ask Depend: Auto Ask STEP 2 Use the table feature to create a table of values. Scrolling through the table shows that y > 0,000 when 0 x 4.2. c The car value exceeds $0,000 for about the first 4.2 years after it is purchased. To check the solution s reasonableness, note that y ø 0,440 when x 5 4 and y ø 8874 when x 5 5. So, 4 < x < 5, which agrees with the solution obtained above. METHOD 2 Use a graph Graph y 5 20,000(0.85) x and y 5 0,000 in the same viewing window. Set the viewing window to show 0 x 8 and 0 y 25,000. Using the intersect feature, you can determine that the graphs intersect when x ø X X=4.2 Y Intersection X= Y=0000 The graph of y 5 20,000(0.85) x is above the graph of y 5 0,000 when 0 x < c The car value exceeds $0,000 for about the first 4.27 years after it is purchased. 526 Chapter 7 Exponential and Logarithmic Functions

13 E XAMPLE 2 Solve a logarithmic inequality Solve log 2 x 2. Solution METHOD Use a table STEP Enter the function y 5 log 2 x into a graphing calculator as y 5} log x. log 2 Y=log(X)/log(2) Y2= Y3= Y4= Y5= Y6= STEP 2 Use the table feature to create a table of values. Identify the x-values for which y 2. These x-values are given by 0 < x 4. Make sure that the x-values are reasonable and in the domain of the function (x > 0). c The solution is 0 < x 4. X X=4 Y METHOD 2 Use a graph Graph y 5 log 2 x and y 5 2 in the same viewing window. Using the intersect feature, you can determine that the graphs intersect when x 5 4. The graph of y 5 log 2 x is on or below the graph of y 5 2 when 0 < x 4. Intersection X=4 Y=2 c The solution is 0 < x 4. PRACTICE EXAMPLE on p. 526 for Exs. 6 Solve the exponential inequality using a table and using a graph.. 3 x } 3 2 x > (0.35) x (0.96) x < (.6) x } 7 2 x > 235 EXAMPLE 2 on p. 527 for Exs. 7 2 Solve the logarithmic inequality using a table and using a graph. 7. log 3 x 3 8. log 5 x < 2 9. log 6 x log 4 x 2 > log 2 x > log 7 x 3. FINANCE You deposit $000 in an account that pays 3.5% annual interest compounded monthly. When is your balance at least $200? 4. RATES OF RETURN An investment that earns a rate of return r doubles in value in t years, where t 5} ln 2 and r is expressed as a decimal. What ln ( r) rates of return will double the value of an investment in less than 0 years? Extension: Solve Exponential and Logarithmic Inequalities 527

14 Investigating g Algebra ACTIVITY Use before Lesson Model Data with an Exponential Function MATERIALS 00 pennies cup graphing calculator TEKS TEXAS classzone.com Keystrokes a.5, a.6, 2A..B, 2A..F QUESTION How can you model data with an exponential function? E XPLORE Collect and record data STEP Make a table Make a table like the one shown to record your results. Number of toss, x Number of pennies remaining, y???????? STEP 2 Perform an experiment Record the initial number of pennies in the table, and place the pennies in a cup. Shake the pennies, and then spill them onto a flat surface. Remove all of the pennies showing heads. Count the number of pennies remaining, and record this number in the table. STEP 3 Continue collecting data Repeat Step 2 with the remaining pennies until there are no pennies left to return to the cup. DRAW CONCLUSIONS Use your observations to complete these exercises. What is the initial number of pennies? By what percent would you expect the number of pennies remaining to decrease after each toss? 2. Use your answers from Exercise to write an exponential function that should model the data in the table. 3. Use a graphing calculator to make a scatter plot of the data pairs (x, y). In the same viewing window, graph your function from Exercise 2. Is the function a good model for the data? Explain. 4. Use the calculator s exponential regression feature to find an exponential function that models the data. Compare this function with the function you wrote in Exercise Chapter 7 Exponential and Logarithmic Functions