16.2 Solving Exponential Equations

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1 Name Class Date 16.2 Solving Exponential Equations Essential Question: What are some ways you can solve an equation of the form ab x = c, where a and c are nonzero real numbers and b is greater than 0 and not equal to 1? Resource Locker Explore Solving Exponential Equations Graphically One way to solve exponential equations is graphically. First, graph each side of the equation separately. The point(s) at which the two graphs intersect are the solutions of the equation. First, look at the equation 275 e 0.06x = To solve the equation graphically, split it into two separate equations. y 1 = y 2 = B What will the graphs of y 1 and y 2 look like? Graph y 1 and y 2 using a graphing calculator. The x-coordinate of the point of intersection is approximately. So, the solution of the equation is x. Now, look at the equation 102x 4 = 10. Split the equation into two separate equations. y 1 = y 2 = Module Lesson 2

2 What will the graphs of y 1 and y 2 look like? Graph y 1 and y 2 using a graphing calculator. The x-coordinate of the point of intersection is. So, the solution of the equation is x. Reflect 1. How can you check the solution of an exponential equation after it is found graphically? Explain 1 Solving Exponential Equations Algebraically In addition to solving exponential equations graphically, exponential equations can be solved algebraically. One way to solve exponential equations is to rewrite them in logarithmic form. Another way is to use the Property of Equality for Logarithmic Equations which states that for any positive numbers x, y, and b (b 1), log b x = log b y if and only if x = y. Example 1 10 = 5 e 4x Solve the equations. Give the exact solution and an approximate solution to three decimal places. 10 = 5 e 4x Original equation 2 = e 4x Divide both sides by 5. ln 2 = 4x Rewrite in logarithmic form. ln 2 4 = 4x Divide both sides by 4. 4 ln 2 4 = x Simplify x Evaluate. Round to three decimal places. Module Lesson 2

3 B 5 x - 4 = 7 5 x - 4 = 7 Original equation 5 x = 7 + Add to both sides. 5 x = Simplify. log 5 x = log Take the common logarithm of both sides. = log 11 Power Property of Logarithms log x = _ Divide both sides by log 5. log x Evaluate. Round to three decimal palces. Reflect 2. Consider the equation 2 x - 3 = 85. How can you solve this equation using logarithm base 2? 3. Discussion When solving an exponential equation with base e, what is the benefit of taking the natural logarithm of both sides of the equation? Your Turn Solve the equations. Give the exact solution and an approximate solution to three decimal places e x = x = 12 Module Lesson 2

4 Explain 2 Solve a Real-World Problem by Solving an Exponential Equation n Suppose that $250 is deposited into an account that pays 4.5% compounded quarterly. The equation A = P (1 + r gives the amount A in the account after n quarters for an initial investment P that earns interest at a rate r. Solve for n to find how long it will take for the account to contain at least $500. Analyze Information Identify the important information. The initial investment P is $. The interest rate is %, so r is. The amount A in the account after n quarters is $. 4 ) Formulate a Plan Solve the equation for A = P (1 + information and using logarithms. 4) n r for by substituting in the known Solve ) n = (1 + _ 4 n = ( 1 + _ ) Substitute. Divide both sides by = n Evaluate the expression in parentheses. log 2 = log n Take the common logarithm of both sides. log log log 2 = log Power Property of Logarithms = n Divide both sides by log n Evaluate. Module Lesson 2

5 Justify and Evaluate It will take about quarters, or about years, for the account to contain at least $500. Check by substituting this value for n in the equation and solving for A. A = 250 ( = 250 ( ) _ 4 ) Substitute. Evaluate the expression in parentheses. 250 ( ) Evaluate the exponent. Multiply. So, the answer is reasonable. Your Turn 6. How long will it take to triple a $250 initial investment in an account that pays 4.5% compounded quarterly? Elaborate 7. Describe how to solve an exponential equation graphically. 8. Essential Question Check-In Describe how to solve an exponential equation algebraically. Module Lesson 2

6 Evaluate: Homework and Practice Solve the equations graphically e 0.1x = 60 Online Homework Hints and Help Extra Practice e 2x = 75 e 3x 3. 5 = 625 e 0.02x Module Lesson 2

7 Solve the equations graphically. Then check your solutions algebraically e 6x = 5 e -3x e 0.4x = 2000 Solve the equations. Give the exact solution and an approximate solution to three decimal places x = e 3x = x = e _ 2x = 250 Module Lesson 2

8 10. (1 0 x ) 2 x_ + 90 = = 30 Solve. 12. The price P of a gallon of gas after t years is given by the equation P = P 0 ( 1 + r ) t, where P 0 is the initial price of gas and r is the rate of inflation. If the price of a gallon of gas is currently $3.25, how long will it take for the price to rise to $4.00 if the rate of inflation is 10.5%? 13. Finance The amount A in a bank account after t years is given by the equation A = A 0 (1 + r_ 6 ) 6t, where A 0 is the initial amount and r is the interest rate. Suppose there is $600 in the account. If the interest rate is 4%, after how many years will the amount triple? Image Credits: Kabby/ Shutterstock Module Lesson 2

9 14. A baseball player has a 25% chance of hitting a home run during a game. For how many games will the probability of hitting a home run in every game drop to 5%? 15. Meteorology In one part of the atmosphere where the temperature is a constant -70 F, pressure can be expressed as a function of altitude by the equation P (h) = 128 (10) h, where P is the atmospheric pressure in kilopascals (kpa) and h is the altitude in kilometers above sea level. The pressure ranges from 2.55 kpa to 22.9 kpa in this region. What is the range of altitudes? Module Lesson 2

10 16. You can choose a prize of either a $20,000 car or one penny on the first day, double that (2 cents) on the second day, and so on for a month. On what day would you receive at least the value of the car? 17. Population The population of a small coastal resort town, currently 3400, grows at a rate of 3% per year. This growth can be expressed by the exponential equation P = 3400 ( ) t, where P is the population after t years. Find the number of years it will take for the population to reach 10, A veterinarian has instructed Harrison to give his 75-lb dog one 325-mg aspirin tablet for arthritis. The amount of aspirin A remaining in the dog s body after t minutes can be expressed by A = 325 ( 1 2 ) t 15. How long will it take for the amount of aspirin to drop to 50 mg? Image Credits: (t) Destinations/Corbis; (b) Birgid Allig/Corbis Module Lesson 2

11 19. Agriculture The number of farms in Iowa (in thousands) can be modeled by N (t) = 119 (0.987) t, where t is the number of years since According to the model, when will the number of farms in Iowa be about 80,000? 20. Match the equations with the solutions. a. 9 e 3x = 27 x b. 9 e x = 27 x c. 9 e 3x - 4 = 27 x d. 9 e 3x + 2 = 27 x Module Lesson 2

12 H.O.T. Focus on Higher Order Thinking 21. Explain the Error A student solved the equation e 4x - 6 = 10 as shown. Find and correct the student s mistake. Is there an easier way to solve the problem? Verify that both methods result in the same answer. e 4x - 6 = 10 e 4x = 16 log e 4x = log 16 4x log e = log 16 4x (1) = log 16 x = _ log 16 4 x Multi-Step The amount A in an account after t years is given by the equation A = P e rt, where P is the initial amount and r is the interest rate. a. Find an equation that models approximately how long it will take for the initial amount P in the account to double with the interest rate r. Write the equation in terms of the interest rate expressed as a percent. Module Lesson 2

13 b. The Rule of 72 states that you can find the approximate time it will take to double your money by dividing 72 by the interest rate. The rule uses 72 instead of 69 because 72 has more divisors, making it easier to calculate mentally. Use the Rule of 72 to find the approximate time it takes to double an initial investment of $300 with an interest rate of 3.75%. Determine that this result is reasonable by solving the equation A = P 0 (1.0375) t, where A is the amount after t years and P 0 is the initial investment. 23. Represent Real-World Problems Suppose you have an initial mass M 0 of a radioactive substance with a half-life of h. Then the mass of the parent isotopes at time t is P (t) = M 0 ( 1 2 ) t h. Since the substance is decaying from the original parent isotopes into the new daughter isotopes while the mass of all the isotopes remains constant, the mass of the daughter isotopes at time t is D (t) = M 0 - P (t). Find when the masses of the parent isotopes and daughter isotopes are equal. Explain the meaning of your answer and why it makes sense. Module Lesson 2

14 Lesson Performance Task The frequency of a note on the piano, in Hz, is related to its n position on the keyboard by the function f (n) = , where n is the number of keys above or below the note concert A, concert A being the A key above middle C on the piano. Using this function, find the position n of the key that has a frequency of 110 Hz. Why is this number a negative value? Ebby May/Getty Images Module Lesson 2