ࡱ> npm#` bjbj\.\. >D>D/GLLL6668n$w:ɤɤɤ$ch9L@ɤ18L4L P*66G0w:$4L ɤv?TDץɤɤɤ$ɤɤɤw$$$<ra:$$$ra`r Lesson Title: Analyzing the Relationship between a Logarithmic Function and Its Inverse Lesson Summary: This activity will give students the opportunity to investigate the inverse of the logarithmic function and explore several uses of the logarithmic function and its inverse in real world applications. The student will determine numerous ordered pairs that satisfy a given logarithmic function, reverse the x and y coordinates, and use regression to determine the function that represents the inverse of the given logarithmic function. The student will then use this relationship to solve problems. Key Words: inverse, families of functions (quadratic, logarithmic, exponential), regression analysis, best fit Background knowledge: The student will know how to find the inverse of a linear or quadratic function algebraically, graphically, and numerically. The students will be able to recognize the graphs of polynomial and exponential functions. The student will have been introduced to algebraic properties of logarithms. The student should have a general knowledge of graphing, table usage, and regression analysis with a graphing calculator. OAC Standard(s) Addressed: Patterns, Functions, and Algebra Benchmarks: 8-10 C, D, E; 11-12 A Grade Level Indicators: 10.10, 11.3, 11.5, 11.6 Learning Objectives: The student should demonstrate numerically and graphically that the inverse of a logarithmic function is an exponential function. The student should solve logarithmic and exponential equations using inverse operations, and recognize real world applications that can be modeled with these functions. Materials: Activity handout, worksheet, and graphing calculator Suggested procedures: The lesson will be introduced with a teaser regarding a method of measuring the volume at which the human eardrum will rupture. Students should be grouped in pairs, but each should enter and record data individually. Assessment(s): Formative assessment will consist of the questions and extensions in the activity and the class discussion following its completion. Summative assessment questions should include identifying graphs of exponential and logarithmic functions and using a logarithmic scale (like decibels or the pH scale) to convert data. Activity: Analyzing the Relationship between a Logarithmic Function and Its Inverse This activity will be done after studying algebraic functions, their inverses and basic properties of exponential and logarithmic functions. PART 1: INVESTIGATING A LOGARITHMIC FUNCTION Consider the following function: f(x) = log(x) Make a graph of f(x) using your graphing calculator. The table below lists a series of nine output values from f(x). Fill in the table by using the graph and the TABLE to estimate the missing x values. TABLE 1 XF(x).05.33.891.21.51.752.12.32.5 Using a table of y-values to find the corresponding x-value:  Put  EMBED Equation.3  into Y1 In TBLSET, let TblStart = 0 and "Tbl = 10. As you can see, 1.75 is the y-value somewhere between x = 50 and x = 60. Now, let TblStart = 55 and "Tbl = 1. You can see that 1.75 is a y-value between x = 56 and x = 57. Let TblStart = 56 and "Tbl = 0.1. Once you can be accurate to the tenths place, choose the x-value. Create two lists L1 and L2 in the calculator to store the x and f(x) values. Put the x values in L1 and the f(x) values in L2. Generate a scatter plot of (x, f(x)) using the STAT PLOT feature and the two lists, L1 and L2. Do your x values appear to accurately represent f(x)? If not, modify the values so that they better represent the curve. Generate a new STAT PLOT using L2 and L1 instead of L1 and L2. Hypothesize at least two different families of functions to which this new curve might belong: family 1: __________________________ family 2: __________________________ Using L2 and L1, test your hypothesis with the different regression analysis options and record your results, including the regression equations and R2 values, below. Which of the above selections do you think represents the best fit? Why? If you selected Exponential, you were CORRECT! Congratulate yourself. Look at the exponential equation that you generated. Using your mathematical knowledge and some rounding, estimate g(x), the actual equation of the inverse of f(x): PART 2: Investigating the Inverse Consider the function h(x) = x3. Solve for x if h(x) = 8. What is i(x), the inverse of h(x)? Will i(x) give all the solutions to h(x) = y for any value of y? Explain your answer. Consider the function h(x) = x2. Solve for x if h(x) = 9. Consider the domain of h(x). What would the range of the inverse be? Is there a function, i(x), the inverse of h(x)? Notice that  EMBED Equation.3 will not give all the solutions to h(x)=9. Explain why  EMBED Equation.3  is not the inverse of h(x). Think about the relationship between a function and its inverse. Use that relationship to solve the following: Given: f(x) = log(x). If f(x) = 1.2, find x. Given: g(x) = 10x. If g(x) = 37, find x. part 3: An Application of Logarithms: Hearing and Decibels The decibel is the unit used to measure the intensity of a sound. The decibel is named after Alexander Graham Bell who did much work in the area of sound and loudness. Bell discovered that to obtain a sound that seemed twice as loud as another sound, the intensity (how much sound energy per unit area per second hits the eardrum) of the sound must be multiplied by 10. We call this apparent loudness the "intensity level." The following equation is used to compute the intensity of sound: dB = 10 log (I / Io) where dB represents decibels, I is the intensity of the sound in question (measured in joules per second per square meter), and Io (read as "I naught" ) is the softest sound the human ear can distinguish, being 10-12 joules per second per square meter. Note that "joules per second per square meter" can be expressed more simply as "Watts per square meter" and is written as "W/m2." What exactly does "10-12 joules per second per square meter" (or, more simply, 10-12 W/m2) mean? This intensity corresponds to a sound which will displace particles of air by a mere one-billionth of a centimeter. The human ear can detect such a sound! This faintest sound which the human ear can detect is known as the "threshold of hearing" (TOH). The most intense sound which the ear can safely detect without suffering any physical damage is more than one billion times more intense than the threshold of hearing! It is for this reason that a logarithmic equation is used to measure the intensity of sound. Logarithmic scales are often used when a range of values is extremely large, which is true in the case of the range of intensities that the human ear can detect. Consider this: the human ear can detect a sound as soft as a whisper to as loud as a jet engine flying overhead. That's a big difference in intensity! (Source: Sonic Booms and Logarithms, Robin A. Ward, California Polytechnic State University-San Luis Obispo) Exercises: Complete the worksheet below. Decibels and Logarithms Names:_______________________________________________ Directions: Complete the table using your knowledge of logarithms and the formula:  EMBED Equation.3 , where I = Intensity Sound SourceIntensity (W/m2)Intensity Level (dB)# Times greater than TOHThreshold of hearing (TOH) EMBED Equation.3 Rustling leaves EMBED Equation.3 10Whisper EMBED Equation.3 20Normal conversation60Busy street traffic70Vacuum cleaner80Hearing damage possible EMBED Equation.3 85Lawn mower EMBED Equation.3 Front row at a concert EMBED Equation.3 110Thunderclap (near) EMBED Equation.3 Threshold of pain EMBED Equation.3 130Jet take-off EMBED Equation.3 140Shotgun EMBED Equation.3 Instant perforation of ear drum EMBED Equation.3 160 (Source: Sonic Booms and Logarithms, Robin A. Ward, California Polytechnic State University-San Luis Obispo) Summarize the concepts that were covered in this activity in your own words. Include a discussion about the inverse functions in Parts 1 and 2, and the reading and exercises involving decibels. Put a star next to any concept that was new to you. Extension: Given the formula  EMBED Equation.3  , where [H+ ]is the concentration (in moles per liter) of hydrogen ions: Cindy the saboteur wants to make the pool close so that she wont have to go to swimming practice tomorrow morning. For the pool to remain open, the pH level must be between 7.2 and 7.8. After adding her concoction to the pool, Cindy was able to get a concentration of hydrogen ions of 7.5 10-8 moles per liter. Did she succeed in making the pool unsafe enough to close? Given the formula for Richter Scale,  EMBED Equation.3 , where A is the measured intensity of the earthquake, A0 is the reference intensity, and M is the Richter Scale reading, find the following: Nancy is a scientist who measured the intensity of an earthquake to be 121,000 times the reference intensity. If Nancy needs to report a Richter scale reading to Daniel, a newspaper reporter, what number should Nancy tell Daniel? Find another application for exponential functions or logarithms that was not mentioned in this lesson. What is the base of the function? Be prepared to share a brief explanation of how they are used to the class. SOLUTIONS TABLE 1 x values should be approximately as follows: x1.122.147.715.831.756.3125.9202316.2f(x) .05.33.891.21.51.752.12.32.5 EXERCISES 3-7  (L1,L2) Quadratic Exponential (L2,L1) EMBED Equation.DSMT4  EMBED Equation.DSMT4  9.  EMBED Equation.3  10. x = 2 11.  EMBED Equation.3  or  EMBED Equation.3  12. Yes, because  EMBED Equation.3  is defined for all possible outputs of  EMBED Equation.3 . In other words, both functions are defined for all real numbers. 13.  EMBED Equation.3  14. No, if you let  EMBED Equation.3 , then  EMBED Equation.3 , and so  EMBED Equation.3  is only the inverse of  EMBED Equation.3  for  EMBED Equation.3 . 15. The inputs (x-values) of  EMBED Equation.3  can be positive or negative, but the outputs (y-values) of  EMBED Equation.3  are only non-negative. The xs and ys of these two functions cannot always be swapped to get the other function. 16. x = 15.85 17. x = 1.568 Decibels and Logarithms Names:_______________________________________________ Directions: Complete the table using your knowledge of logarithms and the formula:  EMBED Equation.3 , where I = Intensity Sound SourceIntensity (W/m2)Intensity Level (dB)# Times greater than TOHThreshold of hearing (TOH) EMBED Equation.3 00Rustling leaves EMBED Equation.3 1010Whisper EMBED Equation.3 20100Normal conversation10-6601,000,000Busy street traffic10-77010,000,000Vacuum cleaner10-480100,000,000Hearing damage possible EMBED Equation.3 85316,000,000Lawn mower EMBED Equation.3 90109Front row at a concert EMBED Equation.3 1101011Thunderclap (near) EMBED Equation.3 1201012Threshold of pain EMBED Equation.3 1301013Jet take-off EMBED Equation.3 1401014Shotgun EMBED Equation.3 1501015Instant perforation of ear drum EMBED Equation.3 1601016 * * - Note that 1016 is equal to 10,000,000,000,000,000 or 10 quadrillion. Extension pH = log[7.5 x 10-8] = 7.12 Yes, the saboteur succeeded, the pool was unsafe. M = log(121,000) = 5.08 Answers will vary. This is desirable so that students do research and find things to add to the discussion that you may not have found or thought of. 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