16.2 Solving Exponential Equations

Transcription

1 Locker LESSON 16.2 Solving Exponential Equations Texas Math Standards The student is expected to: A2.5.D Solve exponential equations of the form y = ab x where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions. Mathematical Processes A2.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Language Objective 1.A, 2.I.4, 3.E, 3.H.3, 4.D, 4.F Have students work with a partner to describe the steps for solving linear, exponential, and now logarithmic equations algebraically and graphically. 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? A2.5.D Solve exponential equations of the form y = ab x where a is a nonzero real number and b is greater than zero and not equal to one Explore Solving Exponential Equations Graphically Resource Locker 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 = 275 e 0.06x y 2 = 1000 What will the graphs of y 1 and y 2 look like? The graph of y 1 will be increasing exponentially, and the graph of y 2 will be a horizontal line. Graph y 1 and y 2 using a graphing calculator. ENGAGE 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? You can graph both sides of the equation and look for the point of intersection, or you can take the logarithm of both sides and solve algebraically. The x-coordinate of the point of intersection is approximately So, the solution of the equation is x Now, look at the equation 10 2x = Split the equation into two separate equations. 10 2x y 1 = 4 y 2 = 10 PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the frequencies of piano notes are related by an exponential function rather than a linear function. Then preview the Lesson Performance Task. Module Lesson 2 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? A2.5.D Solve exponential equations of the form y = ab x where a is a nonzero real number and b is greater than zero and not equal to one Explore Solving Exponential Equations Graphically Resource 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 = 275 e 0.06x 1000 What will the graphs of y 1 and y 2 look like? The graph of y 1 will be increasing exponentially, and the graph of y 2 will be a horizontal line. 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 10 2x = Split the equation into two separate equations. y 1 = 10 2x HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. y 2 = 10 4 Module Lesson Lesson 16.2

2 What will the graphs of y 1 and y 2 look like? The graph of y 1 will be increasing exponentially, and the graph of y 2 will be a horizontal line. Graph y 1 and y 2 using a graphing calculator. EXPLORE Solving Exponential Equations Graphically INTEGRATE TECHNOLOGY Students have the option of completing the Explore activity either in the book or online. The x-coordinate of the point of intersection is 2. So, the solution of the equation is x 2. Reflect 1. How can you check the solution of an exponential equation after it is found graphically? Substitute the solution in for the value of x in the equation and use a calculator to evaluate. Explain 1 Solving Exponential Equations Algebraically In addition to solving exponential equations graphically, exponential equations can be solved algebraically. The Property of Equality for Logarithmic Equations states that for any positive numbers x, y, and b, (b 1), log b x = log b y if and only if w = 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 = ln e 4x ln 2 = 4x ln e ln 2 = 4x Simplify ln e. Take the natural logarithm of both sides. Power Property of Logarithms ln 2 4 = 4x Divide both sides by 4. 4 ln 2 4 = x Simplify x Evaluate. Round to three decimal palces. Module Lesson 2 PROFESSIONAL DEVELOPMENT Learning Progressions In previous lessons, students solved exponential equations in which both sides of the equation could be written as powers with the same base. Now that students have learned about logarithms, they can solve exponential equations in which the bases are not the same. Students will solve these exponential equations both graphically and algebraically by taking the logarithm of both sides. Students should notice the similarities between the Property of Equality for Logarithmic Equations and the Property of Equality for Exponential Equations. QUESTIONING STRATEGIES Why would there be no solution if the right side of the equation were a number less than or equal to 0? The graph of the left side of the equation has a horizontal asymptote at the x-axis, so there would be no intersection point for the two functions and no value of the domain for which both functions have the same value. EXPLAIN 1 Solving Exponential Equations Algebraically QUESTIONING STRATEGIES How do you use the Property of Equality for Exponential Equations to solve the equation 2 6x x + 1 = 8? Write both sides of the equation as powers of 2, then set the exponents equal, and solve for x. 6x 3 x = (2 ) 6x = 3x + 3 x = 1 Why is it not possible to use the Property of Equality for Exponential Equations to solve the equation 2 6x = 10 x + 1? because 2 and 10 cannot be written as powers of the same base How does the Property of Equality for Logarithmic Equations enable you to solve the equation 2 6x = 10 x + 1? The bases of the powers do not need to be the same. You can take the log of both sides. Solving Exponential Equations 884

3 INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Discuss with students how the Property of Equality for Logarithmic Equations can be verified by applying the definition of a logarithm to the equation log b x = log b y, rewriting it in exponential form. If log b x = log b y, then b log b y = x. B 5 x - 4 = 7 5 x - 4 = 7 Original equation 5 x = Add 4 to both sides. 5 x = 11 Simplify. log 5 x = log 11 Take the common logarithm of both sides. x log 5 = log 11 Power Property of Logarithms log 11 x = _ Divide both sides by log 5. log 5 By the Definition-Based Properties of Logarithms, b log b y = y. Therefore, x = y. AVOID COMMON ERRORS Some students may apply the Property of Equality for Logarithmic Equations incorrectly, taking the log of each term of an equation. Correct this error, instructing students to try to transform the equation so that there is only one term on each side of the equal sign. At that point, they can take the log of the expression on each side of the equation. Reflect x Evaluate. Round to three decimal palces. 2. Consider the equation 2 x - 3 = 85. How can you solve this equation using logarithm base 2? Take the logarithm base 2 of both sides. Then rewrite log 2 85 using the Change of Base Property of Logarithms so the exact solution can be found using a scientific calculator. 2 x - 3 = 85 log 2 2 x - 3 = log 2 85 (x - 3) log 2 2 = log 2 85 x - 3 = log 2 85 x = log log 85 x = log x Discussion When solving an exponential equation with base e, what is the benefit of taking the natural logarithm of both sides of the equation? Taking the natural logarithm of both sides will result in ln e on one side, which simplifies to 1. Module Lesson 2 COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs. Provide each pair with identical sets of several different exponential equations, and ask them to solve each equation two different ways. Have pairs share their solutions with the class, comparing their work with that of the other pairs of students. Discuss the results. 885 Lesson 16.2

4 Your Turn Solve the equations. Give the exact solution and an approximate solution to three decimal places e x = x = 12 2 e x - 1 = 75 Explain 2 e x - 1 = 37.5 ln e x - 1 = ln 37.5 (x - 1) (ln e) = ln 37.5 (x - 1) (1) = ln 37.5 x = ln log 6 3x = log 12 3x log 6 = log 12 log 12 x = 3 log 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 4 ) 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. EXPLAIN 2 Solving a Real-World Problem by Solving an Exponential Equation QUESTIONING STRATEGIES When working with a formula that models a real-world situation, how can you tell whether you will need to take the log of both sides of the equation? You need to take the log of both sides of the equation if the variable for which you are solving is in the exponent. Analyze Information Identify the important information. The initial investment P is $ 250. The interest rate is 4.5 %, so r is The amount A in the account after n quarters is $ 500. Formulate a Plan n Solve the equation for A = P (1 + r for information and using logarithms. 4) n by substituting in the known Module Lesson 2 DIFFERENTIATE INSTRUCTION Multiple Representations Some students may suggest using the definition of a logarithm to solve an equation such as 5 x + 3 = 12, rewriting the equation in logarithmic form, and using the Change of Base Property to solve for x. x + 3 = log 5 12 log 12 x = log Students should be encouraged to use this alternate method when possible, if they prefer. Show them how this leads to the same result as taking the log of both sides. Solving Exponential Equations 886

5 INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Discuss with students how the formula for compound interest changes for different compounding periods. Help them to see that the value of A increases as the number of compounding periods per year increases. Show how this concept leads to the formula for situations in which the interest is compounded continuously. Solve n = 250 (1 + _) 4 2 n = ( 1 + _ ) Substitute. Divide both sides by 250. n 2 = Evaluate the expression in parentheses. log 2 = log n Take the common logarithm of both sides. log 2 = n log Power Property of Logarithms log 2 = n Divide both sides by log log n Evaluate. Justify and Evaluate It will take about quarters, or about 15.5 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 ( _ ) Substitute. = 250 ( ) Evaluate the expression in parentheses ( ) Evaluate the exponent. 500 Multiply. So, the answer is reasonable. Module Lesson 2 LANGUAGE SUPPORT Communicating Math Have students work in pairs to first discuss and then fill in a chart like the one below. Tell students the blank row is to be completed once they have solved logarithmic equations, in the next lesson. Equation linear Algebraic solution Graphical solution Summary of similarities and differences exponential 887 Lesson 16.2 logarithmic

6 Your Turn 6. How long will it take to triple a $250 initial investment in an account that pays 4.5% compounded quarterly? P = 250; A = 3P = 750; r = = 250 ( ) n 3 = ( ) n log 3 = log n log 3 = n log log 3 n = log It will take about 98.2 quarters, or about 24.6 years, to triple. Elaborate 7. Describe how to solve an exponential equation graphically. An exponential equation can be solved graphically by graphing the two sides of the equation and determining where they intersect. 8. Essential Question Check-In Describe how to solve an exponential equation algebraically. An exponential equation can be solved algebraically by taking the logarithm of both sides of the equation and using the properties of logarithms to evaluate. ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Technology Discuss with students how to determine an appropriate viewing window when solving an exponential equation graphically. Discuss how the constants and coefficients in the equation can be used as a guide. QUESTIONING STRATEGIES How does the Property of Equality for Logarithmic Equations make it possible to solve an exponential equation? It moves the variable out of the exponent so it can be isolated. SUMMARIZE THE LESSON What are the steps for solving an exponential equation algebraically? Transform the equation so that there is only one term on each side of the equal sign. Then take the log (or the ln) of both sides of the equation. This moves the variable out of the exponent(s). Then solve the resulting equation for the variable. Module Lesson 2 Solving Exponential Equations 888

7 EVALUATE Evaluate: Homework and Practice Solve the equations graphically e 0.1x = 60 Online Homework Hints and Help Extra Practice ASSIGNMENT GUIDE Concepts and Skills Explore Solving Exponential Equations Graphically Example 1 Solving Exponential Equations Algebraically Example 2 Solving a Real-World Problem by Solving an Exponential Equation Practice Exercises 1 6 Exercises 7 12 Exercises The x-coordinate of the point where the graphs intersect is So, the solution is x e 2x = 75 e 3x TECHNOLOGY Have students provide the window they used for graphing each equation. Have them share their strategies for determining the windows with their classmates, so that students who are struggling with this may gain some insight. The x-coordinate of the point at which the graphs intersect is So, the solution is = 625 e 0.02x The x-coordinate of the point at which the graphs intersect is So, the solution is x Module Lesson 2 Exercise Depth of Knowledge (D.O.K.) Mathematical Processes Recall of Information 1.F Analyze relationships Recall of Information 1.C Select tools Skills/Concepts 1.A Everyday life 21 2 Skills/Concepts 1.F Analyze relationships 22 3 Strategic Thinking 1.G Explain and justify arguments Strategic Thinking 1.A Everyday life 889 Lesson 16.2

8 Solve the equations graphically. Then check your solutions algebraically e 6x = 5 e -3x AVOID COMMON ERRORS It is easy to make errors when entering logarithmic expressions on a graphing calculator. Prompt students to be sure to close the parentheses after entering the argument for each log. Have them log 12 practice entering expressions such as 2log 5 + log 3, and checking them for correctness. The x-coordinate of the point at which the graphs intersect is So, the solution is x Check: 10 e 6 (-0.08) -3(-0.08) 5 e 10 e e e 0.24 e e e 0.4x = The answer is reasonable. The x-coordinate of the point at which the graphs intersect is x So, the solution is x Check: 450 e 0.4 (3.73) e The answer is reasonable. Module Lesson 2 Solving Exponential Equations 890

9 INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Encourage students to check their solutions for correctness by substituting the value into the original equation and verifying that it makes the equation true. Point out that this will help them identify any errors, and encourage them to double check that they ve entered the logarithmic expressions correctly in the graphing calculator. Impress upon students that errors working with these expressions are often a result of missing, or incorrect use of, parentheses e 1_ 3 x = 225 e 2_ 3 x The x-coordinate of the point at which the graphs intersect is 2.4. So, the solution is x 2.4. Check: 500 e 1_ 3 (2.4) 225 e 2_ 3 (2.4) 500 e e The answer is reasonable. Solve the equations. Give the exact solution and an approximate solution to three decimal places x = e 3x = x = x - 9 = 7 log 6 3x - 9 = log 7 (3x - 9) log 6 = log 7 3x - 9 = log 7 log 6 x = 1 3 ( log 7 log 6 ) e 3x = 42 e 3x = 6 ln e 3x = ln 6 3x ln e = ln 6 3x (1) = ln x + 2 = e _ 2x = x + 2 = 12 log 1 1 6x + 2 = log 12 (6x + 2) log 11 = log 12 log 12 6x + 2 = log 11 log 12 6x = log 11-2 x = 1 log 12 6 ( log 11 ) x = ln e 2x = 250 ln e 2x = ln 250 ( 2x ) ln e = ln 250 ( 2x )(1) = ln 250 2x - 1 = 3 ln ln x = Module Lesson Lesson 16.2

10 11. (1 0 x ) = x_ 4 = 30 (1 0 x ) = x = 15 log 1 0 2x = log 15 2x log 10 = log 15 2x (1) = log 15 log 15 x = x_ 4 = 30 log 5 x_ 4 = log 30 x_ 4 log 5 = log 30 4 log 30 x = log QUESTIONING STRATEGIES When is it preferable to take the natural log, as opposed to the common log, of both sides of an exponential equation that models a real-world situation? when the base of the power in the model is e Solve. 13. 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%? 4 = 3.25 (1.105) t t log log t log t log log t log It will take about 2 years for the price of gas to rise to $ 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? 1800 = 600 ( ) 6t 3 = ( ) 6t t log 3 log t log 3 6t log log 3 t 6 log The amount in the account will triple after about 27.4 years. Image Credits: Kabby/ Shutterstock Module Lesson 2 Solving Exponential Equations 892

11 15. 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%? Let n be the number of games. P (home run in every game) = n 0.05 = n log 0.05 = log n log 0.05 = n log 0.25 log 0.05 n = log The probability of hitting a home run in every game drops to 5% after about 2 games. 16. 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? Find the altitude for a pressure of 2.55 kpa = 128 (10) h h log log h log h log 10 log h (1) log h Find the altitude for a pressure of 22.9 kpa = 128 (10) h h log log h log h log10 log h (1) log h The altitude ranges from about km to about km above sea level. Module Lesson Lesson 16.2

12 17. 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? $20,000 is 2,000,000 cents. On day 1, you would receive 1 cent, or 2 0 cents. On day 2, you would receive 2 cents, or 2 1 cents, and so on. So, on day n you would receive 2 n - 1 cents. 2 n - 1 = 2,000,000 log 2 n - 1 = log 2,000,000 (n - 1) log 2 = log 2,000,000 log 2,000,000 n - 1 = log 2 log 2,000,000 n = log 2 You would receive at least the value of the car on day 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, ,000 = 3400 ( ) t 2.94 (1.03) t log 2.94 log t log 2.94 t log 1.03 t log 2.94 log It will take about 36.5 years 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 t be expressed by A = ( 2 ). How long will it take for the amount of aspirin to drop to 50 mg? t 2) 50 = 325 ( 1_ = (0.5) t 15 log = log (0.5) t 15 log = t log t = 15 log ( log 0.5 ) 40.5 It will take about 40.5 minutes for the amount of aspirin to drop to 50 mg. Image Credits: (t) Destinations/Corbis; (b) Birgid Allig/Corbis Module Lesson 2 Solving Exponential Equations 894

13 20. 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? 80 = 119 (0.987) t t log log t log t log log t log The number of farms in Iowa will be about 80,000 after about 30 years, or around Match the equations with the solutions. a. 9 e 3x = 27 B x b. 9 e x = 27 D x c. 9 e 3x - 4 = 27 A x d. 9 e 3x + 2 = 27 C x e 3x = 27 e 3x = 3 ln e 3x = ln 3 3x ln e = ln 3 3x (1) = ln 3 x = ln e x = 27 e x = 3 ln e x = ln 3 x ln e = ln 3 x (1) = ln e 3x - 4 = 27 e 3x - 4 = 3 ln e 3x - 4 = ln 3 (3x - 4) ln e = ln 3 (3x - 4) (1) = ln 3 3x = ln x = ln e x + 2 = 27 9 e x = 25 e x = 25 9 ln e x 25 = ln 9 25 x ln e = ln 9 x (1) 25 = ln Module Lesson Lesson 16.2

14 H.O.T. Focus on Higher Order Thinking 22. 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 log 16 x = _ 4 x The student evaluated log e incorrectly. log e is not equal to 1. The student should have evaluated the logarithm correctly or used the natural log. e 4x - 6 = 10 e 4x = 16 log e 4x = log 16 4xlog e = log 16 log 16 x = 4 log e e 4x - 6 = 10 e 4x = 16 ln e 4x = ln 16 4x ln e = ln 16 4x (1) = ln 16 ln 16 x = PEER-TO-PEER DISCUSSION Have students work with a partner to find two different ways of solving the equation 16 x = 32 x - 2 algebraically, one solution without the use of logs, and the other using logs. Have students compare the methods, and discuss their similarities with the class. To solve without using logs, rewrite each expression as a power of 2, set the resulting exponents equal, and solve for x. To solve using logs, take the log of both sides (ideally, using logs of base 2), apply the Power Property of Logarithms, and solve the resulting equation for x. Both methods should lead to the solution 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. 2P = Pe rt 2 = e rt ln 2 = ln e rt ln 2 = rtln e ln 2 = rt (1) t = ln r r t 69 r The equation that models approximately how long it will take for the initial amount to double is t 69 r. Module Lesson 2 Solving Exponential Equations 896

15 JOURNAL Have students describe the methods they have learned for solving exponential equations. 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. t 72_ 3.75 = 19.2 By the Rule of 72, the amount will double in about 19.2 years. Check: 600 = 300 (1.0375) t 2 = t log 2 = log t log 2 = t log log 2 t = log The results are approximately equal, so 19.2 years is reasonable. 24. 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 ) t h. Since the substance is decaying from the original parent 2 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. To find when the masses of the parent isotopes and daughter isotopes are equal, solve D(t) = P(t) D (t) = P (t) M 0 - M 0 ( 1_ 2) t_ h = M 0 ( 1_ 2) t_ h M 0 = 2 M 0 ( 1_ 2) t_ h M 0 t_ = (0.5) h 2 M = (0.5) t_ h log 0.5 = log (0.5) t_ h log 0.5 = t_ log 0.5 h h = t The masses of the parent isotopes and daughter isotopes are equal when t = h. So, the masses of the isotopes will be equal after 1 half-life. This makes sense because after one half-life, the mass of the parent isotopes will be 1_ of the 2 initial mass. Since the parent isotopes decay into the daughter isotopes, the other half is now made of daughter isotopes. So, the two masses are equal. Module Lesson Lesson 16.2

16 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? n 110 = n 0.25 = 2 12 n lo g = lo g = n = n 24 keys to the left, or below, concert A. A negative value for n means that the key is to the left of, or lower on the keyboard than, concert A. AVOID COMMON ERRORS When solving for the exponent of 2 12 n, students may take the square root instead of lo g 2. Explain that you would take the square root if 2 were the exponent: n 2. But, in this problem, 2 is the base, so you need to take n 12 n the log base 2: lo g 2 2 = 12. QUESTIONING STRATEGIES What is the rate of growth for the frequencies of the notes on the piano? The rate of growth is 2 every 12 units. How often does the frequency double? every 12 keys (every 8 diatonic degrees) Ebby May/Getty Images How many keys are in an octave? An octave is two notes whose frequencies are in a ratio of 1 to 2. An octave is divided into 12 notes, or keys, on the keyboard. Module Lesson 2 EXTENSION ACTIVITY Have students research the whole tone scale, a scale that has been used in Western classical music, jazz, and Indian classical music. Have students write the function w (n) that gives the frequencies of the notes in the scale, and then have them compare it to the function for the frequencies of piano notes in the Performance n 6 Task. A function that includes the note concert A is w (n) = The whole tone scale is a six-note scale corresponding to every other note on the piano keyboard. Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Solving Exponential Equations 898