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Name _______________________________________ Date __________________ Class __________________ LESSON 7-5 Problem Solving Exponential and Logarithmic Equations and Inequalities While John and Cody play their favorite video game, John drinks 4 cups of coffee and a cola, and Cody drinks 2 cups of brewed tea and a cup of iced tea. John recalls reading that up to 300 mg of caffeine is considered a moderate level of consumption per day. The rate at which caffeine is eliminated from the bloodstream is about 15% per hour. 1. John wants to know how long it will take for the caffeine in his bloodstream to drop to a moderate level. a. How much caffeine did John consume? Caffeine Content of Some Beverages Beverage Caffeine (mg per serving) Brewed coffee 103 Brewed tea 36 Iced tea 30 Cola 25 ________________________________________ b. Write an equation showing the amount of caffeine in the bloodstream as a function of time. c. How long, to the nearest tenth of an hour, will it take for the caffeine in John’s system to reach a moderate level? 2. a. Cody thinks that it will take at least 8 hours for the level of caffeine in John’s system to drop to the same level of caffeine that Cody consumed. Explain how he can use his graphing calculator to prove that. ________________________________________________________________________________________ b. What equations did Cody enter into his calculator? ________________________________________ c. Sketch the resulting graph. Choose the letter for the best answer. 3. About how long would it take for the level of caffeine in Cody’s system to drop by a factor of 2? 4. If John drank 6 cups of coffee and a cola, about how long would it take for the level of caffeine in his system to drop to a moderate level? A 0.2 hour B 1.6 hours F 0.5 hour C 2.7 hours G 1.6 hours D 4.3 hours H 4.7 hours J 5.3 hour Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-41 Holt Algebra 2 Name _______________________________________ Date __________________ Class __________________ LESSON 7-5 Reading Strategy Use Relationships In solving equations with logarithms and exponents, first use the properties of logarithms and exponential functions to simplify equations. Here are two additional properties that are useful for solving equations. • If x y, then bx by. • If x y, then logb x logb y. Use the equation 2x 16 for Exercise 1. 1. a. Express 16 as a power of 2. b. Rewrite the equation so both sides have the same base. What is the value of x? c. Show how you can check your solution. ________________________________________________________________________________________ Use the equation log10 x 2 for Exercise 2. 2. a. Rewrite the equation using the definition of logarithm. b. What is the solution of the equation? c. What is the value of log 10 100? Use the equation 243x 3x • 92 for Exercise 3. 3. a. Rewrite the equation so that the exponents on both sides have the same base. b. Simplify until it is in the form 3x 3y. c. Solve for x. Use the equation 4x log (10x) 2 2log x 10 for Exercise 4. 4. a. Describe each step in the table to solve the equation. 4x 2log 10x 2log x 10 Use of the Power Property 2 (2x log 10x log x) 10 2x log 10x log x 5 10 x 2x log 5 x b. Simplify and solve the resulting equation. Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-42 Holt Algebra 2 Name _______________________________________ Date __________________ Class __________________ LESSON 7-5 Reteach Exponential and Logarithmic Equations and Inequalities An exponential equation contains an expression that has a variable as an exponent. 5 x 25 is an exponential equation. x 2, since 5 (2) 25. Remember: You can take the logarithm of both sides of an exponential equation. Then use other properties of logarithms to solve. If x y, then log x log y (x 0 and y 0). Solve 6x 2 500. Step 1 Since the variable is in the exponent, take the log of both sides. 6x 2 500 log 6x 2 log 500 Step 2 Use the Power Property of Logarithms: log ap p log a. log 6x 2 log 500 (x 2) log 6 log 500 Step 3 “Bring down” the exponent to multiply. Isolate the variable. Divide both sides by log 6. (x 2) log 6 log 500 x2 Step 4 Solve for x. Subtract 2 from both sides. x Step 5 log 500 log 6 log 500 2 log 6 Use a calculator to approximate x. x 1.468 Step 6 Use a calculator to check. 61.468 2 499.607 Solve and check. 1. 4x 32 2. 34x 90 3. 5x 3 600 log 4x log 32 log 34x log 90 x log 4 log 32 4x log 3 log 90 ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-38 Holt Algebra 2 Name _______________________________________ Date __________________ Class __________________ LESSON 7-5 Reteach Exponential and Logarithmic Equations and Inequalities (continued) A logarithmic equation contains a logarithmic expression that has a variable. log5 x 2 is a logarithmic equation. x 25, since 52 25. Combine and use properties of logarithms to solve logarithmic equations. Solve: log 80x log 4 1 Step 1 Use the Quotient Property of Logarithms. log 80x log 4 1 log Step 2 logx logy log 80 x 1 4 x y Simplify. log 80 x 1 4 log 20x 1 Step 3 Use the definition of the logarithm: if bx a, then logb a x. log10 20x 1 Remember: Use 10 as the base when the base is not given. 101 20x Step 4 Solve for x. Divide both sides by 20. 10 20x 1 x 2 Solve and check. 4. log3 x4 8 4 log3 x 8 log3 x 8 4 5. log 4 log (x 2) 2 6. log 75x log 3 1 log 4 (x 2) 2 log10 (4x 8) 2 4x 8 102 ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-39 Holt Algebra 2 Practice B LESSON 7-5 Exponential and Logarithmic Equations and Inequalities Solve and check. 1. 52x 20 2. 122x 8 15 ________________________ ________________________ 4. 16 64 x7 5x 3. 2x 6 4 0.2x 5. 243 ________________________ 81 x5 ________________________ x 1 8. 32 1 7. 162 2 ________________________ 2x 64 ________________________ ________________________ 6. 25x 125x – 2 ________________________ 1 9. 27 x 6 27 ________________________ Solve. 10. log4 x5 20 11. log3 x6 12 ________________________ 13. log x log 10 14 ________________________ 14. log x log 5 2 ________________________ 16. log (x 4) log 6 1 ________________________ 17. log x log 25 2 2 ________________________ ________________________ 12. log4 (x 6)3 6 ________________________ 15. log (x 9) log (2x 7) ________________________ 18. log (x 1)2 log (5x 1) ________________________ Use a table and graph to solve. 19. 2 x 5 64 ________________________ 20. log x3 12 ________________________ 21. 2x 3x 1296 ________________________ Solve. 22. The population of a small farming community is declining at a rate of 7% per year. The decline can be expressed by the exponential equation P C (1 0.07) t , where P is the population after t years and C is the current population. If the population was 8,500 in 2004, when will the population be less than 6,000? Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-39 Holt Algebra 2