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Foerster_IRB2_023-025 9/24/03 8:27 PM Page 23 Supplementary Lesson: Log-log and Semilog Graph Paper Chapter 7 looks at some elementary functions of algebra, including linear, quadratic, power, exponential, and logarithmic. The following supplementary section on log-log and semilog graph paper is intended for use at the conclusion of this chapter. This section provides teaching materials, explorations, teacher notes, and solutions. You can use this lesson in your regular teaching plan or assign it for individual projects, group investigations, or extra credit. Foerster_IRB2_023-025 9/24/03 8:27 PM Page 25 Instructor’s Commentary Objective Given an exponential, power, or logarithmic function, plot its graph on semilog or log-log graph paper. Class Time 2 days The listing of different kinds of graph paper should help the students keep track of which kind to use for a particular problem. Sometimes students may get an error message when they are doing a regression on their grapher. When the grapher does exponential regression, it re-expresses the data using (x , log y), which is why the grapher is unable to do exponential regression on a data set that includes 0 as one of the y-values. Likewise, power regression re-expresses the data using (log x, log y), so x and y must be positive numbers. Homework Assignment Day 1: Q1–Q10, Problems 1, 3, 5, 7 Exploration Notes Day 2: Problems 9, 11, 13, 14 In Mathematical Models and Log Graphs, students will plot points on semilog and log-log graph paper and make connections to the add–multiply (exponential) and multiply–multiply (power) properties. Allow about 20 minutes to complete. Important Terms and Concepts Semilog graph paper Log-log graph paper Lesson Notes Exponential functions can be very hard to read for points with large y-values and for x-values near the horizontal asymptote. One way of showing the function more clearly is to change the y-scale to log y. This makes the y-values smaller for large y-values and negative for positive values of y near zero. Students may have seen such graphs before. For instance, graphs displaying financial information over a period of time, such as the growth of a mutual fund, often are semilog graphs. In the lesson we see that the exponential function y = 1000(0.65x ) is linear when graphed on semilog paper. Therefore, it follows that if the points plotted on the exponential function were re-expressed as (x, log y), then these re-expressed points would be linear on arithmetic graph paper. Similarly, power functions are linear on log-log paper, and a reexpression of the power data as (log x, log y) would be linear on arithmetic graph paper. It may not be obvious which property a set of data exhibits, so Log-log and Semilog Graph Paper gives students practice in finding which regression best fits the data. The results are then verified by using the appropriate graph paper to show the linear relationship. This exploration should take about 25 minutes to complete. Problem Notes • Problems 1–8 reinforce the skills and concepts in the examples and explorations. • Problems 9–14 demonstrate that the log y versus x is linear for exponential functions and that the log y versus log x is linear for power functions. In Example 1, the data are plotted on log-log paper to demonstrate that the relationship between the variables is a power function. In addition, be sure students note the multiply–multiply relation in the x and y variables. Students will revisit parametric equations and see how to use them to graph the log–log relationship. Example 2 demonstrates how to use semilog paper to graph a multiply–add relationship, which models a logarithmic function. Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Log-log and Semilog Graph Paper Instructor’s Commentary / 25 Foerster_IRB2_026-031 9/24/03 8:28 PM Page 26 Log-log and Semilog Graph Paper Student Lesson In many real-world situations, the y-values of a function span a wide range. For instance, suppose the number of bacteria in a culture at the end of any one hour has dropped to 65% of what it was at the beginning of the hour. If there were initially 1000 million bacteria present, then the number of millions of bacteria, y, is given by the exponential function y = 1000 • 0.65x where x is time in hours. y y 1000 1000 100 500 10 x 0 0 5 10 15 x 1 5 10 15 0 Figure A Figure B Figure A shows the graph of this function for the first 15 hours. It is hard to read the graph for x between 10 and 15, even though the axes have a large vertical dimension. In Figure B the vertical scale has been compressed for larger y-values and expanded for smaller y-values. The spaces represent the logarithms of the y-values. You may already have seen such semilog graph paper in the explorations. On this graph it is just as easy to read y for larger x-values. For exponential functions, this graph paper has the added advantage that the points lie in a straight line! The graph of a power function turns out to be a straight line on log-log paper, on which the spaces in the x-direction also represent logarithms, of the x-values. OBJECTIVE Given an exponential, power, or logarithmic function, plot its graph on semilog or log-log graph paper. 26 / Log-log and Semilog Graph Paper Student Lesson Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Foerster_IRB2_026-031 9/24/03 8:28 PM Page 27 Straight-line Semilog and Log-log Graphs To see why an exponential function has a straight-line graph on semilog paper, it is sufficient to show that log y varies linearly with x. To do this, take the log of both sides of the exponential function equation. y = 1000 • 0.65x log y = log (1000 • 0.65x ) Write “log” in front of both sides. log y = log 1000 + log 0.65x The log of a product equals the sum of the logs. log y = log 1000 + x • log 0.65 The log of a power equals the exponent times the log of the base. log y = 3 D 0.1870…x By calculator N log y varies linearly with x, Q.E.D. Form: y = ax + b The graph in Figure B is a straight line with a negative slope. In Problem 13 of this lesson you’ll prove this property in general for exponential functions. In that problem you’ll also prove that the graph of a power function is a straight line on log-log graph paper. U EXAMPLE 1 Snake Skin Problem: Snakes shed their skins periodically. The area of the skin depends on the length of the snake. Suppose you measured the areas in this table. Length (cm) Area (cm2) 5 10 20 60 100 2.25 9 36 324 900 a. Plot the data on log-log graph paper. What do you notice about the points? b. Show by regression that a power function fits these data. Write its particular equation. y 1000 c. Predict the skin area of an anaconda snake 7 meters long. Surprising? d. Use your equation from part b to plot on your grapher the log of area as a function of log of length. What do you notice about the graph? e. Prove algebraically that log of area is a linear function of log of length. What do you notice about the slope of this graph? Solution a. Let x = length in cm and let y = area in cm2. Figure C shows the graph of area as a function of length on log-log paper. The points lie in a straight line. Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press 100 10 x 1 10 100 1 Figure C Log-log and Semilog Graph Paper Student Lesson / 27 Foerster_IRB2_026-031 9/24/03 8:28 PM Page 28 b. By power regression, r = 1. So a power function fits the data. The particular equation is y = 0.09x 2. c. 7 meters is 700 cm, so y = 0.09(7002) = 44,100 cm2. Surprising! d. Put your grapher in parametric mode. Then enter the equations like this: x = log t y = log (0.09 • t 2) The graph is shown in Figure D. It is a straight line. 3 log y 2 1 1 log x 2 Figure D e. log y = log (0.09x 2) Take the log of both sides. log y = log 0.09 + 2 log x log of a product and log of a power log y = D1.0457… + 2 log x By calculator N log y is a linear function of log x, Q.E.D. Form: log y = a(log x ) + b The slope of the linear graph equals the exponent of x. You can confirm this fact geometrically by picking two points on the graph, measuring the run and the rise with a ruler, and observing that rise run = 2. Figure E shows how you can do this. y 3 4 5 1000 2 100 1 Ruler 10 1 2 3 4 5 x 1 10 100 1 Figure E 28 / Log-log and Semilog Graph Paper Student Lesson V Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Foerster_IRB2_026-031 9/24/03 8:28 PM Page 29 U EXAMPLE 2 a. By regression analysis, show numerically that a logarithmic function fits these x- and y-values better than a linear, exponential, or power function. x y 2 10 50 300 900 2.4 5.0 7.6 10.4 12.2 b. Write the particular equation of the best-fitting logarithmic function. c. Plot the points on multiply-add semilog graph paper. Show that the points lie on a line. d. Calculate y if x is 20. Show that this point lies on the graph in part c. e. Calculate x if y is 15. Solution a. Linear: r = 0.8331… Logarithmic: r = 0.999978… Best fit! Exponential: r = 0.7133… Power: r = 0.9736… b. Equation is y = 1.3067… + 1.6001… ln x. Paste it into the y= menu (without rounding). c. Figure F shows that the points lie in a straight line. You can get multiplyadd semilog graph paper by rotating regular add-multiply paper 90− clockwise. y 15 Regression line fits closely. 10 5 (20, 6.1...) is on the line. x 1 10 100 1000 1 Figure F d. If x = 20, then y = 1.3067… + 1.6001… ln 20 = 6.1005… The point (20, 6.1005…) is on the regression line in Figure F. e. If y = 15, then 15 = 1.3067… + 1.6001… ln x. x ≈ 5204.2… Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Use the solver to find x numerically. V Log-log and Semilog Graph Paper Student Lesson / 29 Foerster_IRB2_026-031 9/24/03 8:28 PM Page 30 DEFINITIONS: Kinds of Graph Paper Arithmetic graph paper (“ordinary” graph paper) has linear scales on both axes. Log-log graph paper has logarithmic scales on both axes. Add-multiply semilog paper has a logarithmic scale on the vertical axis only. Multiply-add semilog paper has a logarithmic scale on the horizontal axis only. Problem Set Do These Quickly 5 min Q1. log 5 + log 7 = log –?– Q2. log 18 D log 3 = log –?– Q3. 2 log 7 = log –?– Q4. log10 10 = –?– Q5. log5 5 = –?– Q6. Logarithmic functions have the —?— D —?— pattern. Q7. What pattern do the y-values of a quadratic function follow for regularly spaced x-values? Q8. Name the kind of sequence: 23, 30, 37, 44, … Q9. x 60 ÷ x 10 = x–?– Q10. Find the area of a right triangle with one leg 3 cm and hypotenuse 5 cm. For the exponential functions in Problems 1 and 2, a. Calculate the y-values for the given values of x. b. Plot the points on add-multiply semilog graph paper. c. Show that the points lie on a straight line. 1. y = 1.5 • 2x ; x = 1, 3, 5, 7, 9 2. y = 800 • 0.6x ; x = 2, 4, 6, 8, 10 For the power functions in Problems 3 and 4, a. Calculate the y-values for the given values of x. b. Plot the points on log-log graph paper. c. Show that the points lie on a straight line. d. Measure the slope with a ruler and show that it equals the exponent in the equation. 30 / Log-log and Semilog Graph Paper Student Lesson 3. y = 700xD1.3; x = 1, 5, 10, 30, 100 4. y = 5x 0.8; x = 1, 6, 10, 40, 100 For the logarithmic functions in Problems 5 and 6, a. Calculate the y-values for the given values of x. b. Plot the points on multiply-add semilog graph paper. c. Show that the points lie on a straight line. 5. y = 2 + 3 ln x; x = 1, 4, 10, 200, 1000 6. y = D1 + 2 log x; x = 3, 8, 20, 100, 500 7. Show that an exponential function graph is not straight on log-log paper by plotting the data from Problem 1 on log-log paper. 8. Show that a power function graph is not straight on semilog paper by plotting the data from Problem 4 on semilog paper. For the exponential functions in Problems 9 and 10, plot on your grapher log y as a function of x. Use suitable x- and y-windows. Sketch the result. 9. y = 5 • 3x 10. y = 20 • 0.8x For the power functions in Problems 11 and 12, plot on your grapher log y as a function of log x. You may use parametric mode as in Example 1. Use suitable x- and y-windows. Sketch the result. 11. y = 90xD2 12. y = 2x 3 13. Proof Problems: a. Prove that for the exponential function y = 5 • 3x, log y is a linear function of x. b. Prove in general that for the exponential function y = ab x, log y is a linear function of x. Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Foerster_IRB2_026-031 9/24/03 8:28 PM Page 31 c. Prove that for the power function y = 2x 3, log y is a linear function of log x. d. Prove in general that for the power function y = ax b, log y is a linear function of log x. e. How can you conclude that in a logarithmic function, y is a linear function of log x? 14. Height-Weight Historical Problem: Before there were calculators to do regression analysis efficiently, log-log and semilog graph paper were used to help determine what kind of function fits a given set of data. The kind of paper that gave the straight-line graph indicated the kind of function to use. Suppose that these average weights have been recorded for humans. Height (in.) Weight (lb) 10 20 30 40 50 60 1.7 8.5 21.5 41.6 69.5 105.7 Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press a. Plot the data on add-multiply semilog paper. Plot it again on log-log paper. Which graph seems to be more nearly a straight line? b. Based on your answers to part a, what kind of function fits the data more closely, exponential or power? c. Find algebraically the particular equation of the function in part b. Use the first and the last data points to find the constants in the equation. (This was the way particular equations were found in the days before calculators had regression analysis built in.) d. Do exponential regression and power regression on the given data. Does the regression analysis confirm your conclusion and equation? How can you tell? e. Predict the weight of a 90-in.-tall giant using your equation from part c and again using the regression equation in part d. How closely do the two answers agree? Log-log and Semilog Graph Paper Student Lesson / 31 Foerster_IRB2_032-040 9/24/03 8:28 PM Page 32 Name: Group Members: Exploration: Log-log and Semilog Graph Paper Date: Objective: Given a table of data with irregularly spaced x-values, find by regression the particular equation of the best-fitting function and show that the graph is a straight line on the appropriate kind of graph paper. For Problems 1–3, use this data. x f (x) 3 4.1 For Problems 4–6, use this data. x g(x) 2 261 7 24 5 87 8 37 9 43 11 136 20 16.5 12 212 70 3.7 100 2.4 1. By regression, find the particular equation of the best-fitting linear, exponential, or power function. 2. Plot the data on this semilog graph paper. What do you notice about the points? f (x) 4. By regression, find the particular equation of the best-fitting linear, exponential, or power function. 5. Plot the data on this log-log graph paper. What do you notice about the points? 1000 g(x) 1000 100 100 10 10 x 1 5 10 15 0 3. Calculate f (0) and f (15). Plot these points on the graph. Do these points lie on the same straight line as the given data points? 32 / Log-log and Semilog Graph Paper Explorations x 1 10 100 1 6. Calculate g(1) and g(50). Plot these points on the graph. Do these points lie on the same straight line as the given data points? Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Foerster_IRB2_032-040 9/24/03 8:28 PM Page 33 Name: Group Members: Exploration: Log-log and Semilog Graph Paper continued For Problems 7–9, use this data. Date: 10. What kind of graph paper gives a straight-line graph for x h(x) 1 5.3 6 11.8 Exponential functions? 10 17.0 Power functions? 13 20.9 18 27.4 Linear functions? 7. By regression, find the particular equation of the best-fitting linear, exponential, or power function. 11. Let y = 13 • 7x. What kind of function is this? How do you tell? 8. Plot the data on this arithmetic graph paper. What do you notice about the points? 12. Take the log of both sides of the equation in Problem 11. By appropriate use of the properties of logs, show that log y varies linearly with x. h(x) 30 13. From the semilog paper in Problem 2, measure to the nearest 0.1 mm the following distances on the vertical scale: 25 y-values Distance, mm Ratio log 2 (3, 4, . . .) 1 to 2 1 to 3 20 1 to 4 1 to 5 15 1 to 6 1 to 7 1 to 8 10 1 to 9 1 to 10 14. Make another column in the table above and record 5 Distance from 1 to 2 (3, 4, …) Distance from 1 to 10 x 5 10 15 20 9. Calculate h(0) and h(20). Plot these points on the graph. Plot a straight line through these points. How does the given data relate to the line? Round the ratios to 2 decimal places. 15. Make another column in the table above and record log 2 (3, 4, …). That is, record log 5 in the 1 to 5 row and so on. Round to 2 decimal places. 16. Based on the table above, what do the distances on the y-scale of semilog paper represent? 17. What did you learn as a result of doing this Exploration that you did not know before? Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Log-log and Semilog Graph Paper Explorations / 33 Foerster_IRB2_032-040 9/24/03 8:28 PM Page 34 Name: Group Members: Exploration: Mathematical Models and Log Graphs Date: Objective: Plot graphs of exponential or power functions on semilog or log-log graph paper, respectively. Coffee Cup Problem: You pour a cup of coffee. Three minutes after you pour it, you find that it is 52.7−F above room temperature. You record its temperature each 2 minutes thereafter, finding the following values: x (min) f (x) (−−F) 3 52.7 5 28.8 7 15.7 9 8.6 11 4.7 3. Exponential functions have the add-multiply property. By exponential regression find the particular equation of the best-fitting exponential function. Write the correlation coefficient. 4. Use your equation to find f (0). Show that the answer agrees with the pattern of the points in Problem 1. If the room temperature was 70−, how hot was the coffee when it was poured? 1. Plot these points on this semilog graph paper. The scale on the y-axis is logarithmic, with the distance from the x-axis representing the logarithm of the y-value. For y between 1 and 10, each space represents 1 unit. For y between 10 and 100, each space represents 10 units, and so forth. Connect the points. If any point does not lie on a straight line, go back and check your work. 5. Put log f (x) in a third list in your grapher. Then do linear regression with x and log f (x). Write the result here. Write the correlation coefficient. f (x) 1000 6. On your grapher, plot the linear function from Problem 5 along with the points from the lists of x and log f (x). Use an x-window of [0, 15] and a y-window of [0, 3]. Sketch the result here. What relationship do you notice between this graph and the graph in Problem 1? 100 10 x 1 5 10 15 0 2. Show in the table that the points have, approximately, the add-multiply property. State the result verbally. “Adding 2 to x multiplies f (x) by about —?—.” 34 / Log-log and Semilog Graph Paper Explorations Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Foerster_IRB2_032-040 9/24/03 8:28 PM Page 35 Name: Group Members: Exploration: Mathematical Models and Log Graphs continued Shark Problem: From great white sharks caught in the past, fishermen find the following weights and lengths. x (ft) g(x) (lb) 5 10 15 20 75 600 2025 4800 7. Plot these values on this log-log graph paper. The vertical and horizontal scales are both logarithmic. Connect the points. If the points do not lie in a straight line, find your mistake. Date: 10. Calculate the weight of an 8-ft-long shark. Plot the point on the graph paper in Problem 7, thus showing that the point is on the line. 11. From fossilized sharks’ teeth, naturalists think there were once great white sharks 100 ft long. Based on your mathematical model, how heavy would such a shark be? Surprising? g (x) 10000 12. Put two more lists in your grapher for the shark data of Problem 7, one for log x and one for log g(x). Do linear regression for log g(x) as a function of log x and write the equation. What evidence do you have that a linear function fits these transformed values exactly? 1000 100 x 10 10 100 13. On your grapher, plot the linear function along with the points from the lists of log x and log g(x). Use an x-window of [0, 2] and a y-window of [1, 4]. Sketch the result. How does the graph compare with the log-log graph in Problem 7? 1 8. Show in the table that the points have the multiplymultiply property. State the result verbally. “Multiplying x by —?— multiplies g(x) by —?—.” 14. What did you learn as a result of doing this Exploration that you did not know before? 9. Power functions have the multiply-multiply property. Do power regression on the given points. Write the particular equation. What evidence do you have that the fit is exact? Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Log-log and Semilog Graph Paper Explorations / 35 Foerster_IRB2_032-040 9/24/03 8:28 PM Page 36 Solutions Problem Set b., c. y Q1. log 35 Q2. log 6 Q3. log 49 Q4. 1 Q5. 1 Q6. multiply-add 1000 Q7. Constant 2nd differences Q8. Arithmetic Q10. 6 cm 100 Q9. x 50 2 1. a. x y 10 1 3 3 12 5 48 7 192 9 768 x 1 5 10 15 0 b., c. 3. a. x y y 1 1000 100 700 5 86.3847… 10 35.0831… 30 8.4108… 100 1.7583… b., c. y 1000 10 100 x 1 5 10 15 0 2. a. x y 2 288 4 103.68 6 37.3248 8 13.436928 10 4.83729408 10 x 1 1 10 100 1000 d. Slope = D1.3 36 / Log-log and Semilog Graph Paper Solutions Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Foerster_IRB2_032-040 9/24/03 8:28 PM Page 37 4. a. x y 1 b., c. 5 6 y 20.9648… 10 31.5478… 40 95.6352... 100 199.0535… 10 b., c. y 1000 x 0 1 100 7. 10 100 10 100 1000 y 1000 10 100 x 1 10 1 100 1000 10 d. Slope = 0.8 5. a. x y 1 2 4 6.1588… 10 8.9077… 200 17.8949… 1000 22.7232… x 1 1 8. 1000 y 1000 b., c. y 30 100 20 10 10 x 0 10 1 6. a. x 100 1000 y 3 D0.0457… 8 0.8061… 20 1.6020… 100 3 500 4.3979… Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press x 1 0 25 50 75 100 Log-log and Semilog Graph Paper Solutions / 37 Foerster_IRB2_032-040 9/24/03 8:28 PM Page 38 9. Log-log: log y y 1000 x 1 1 100 10. log y x 1 10 1 11. log y x 1 1 10 100 1000 Log-log gives more nearly a straight line. log x 1 b. Power. 1 c. a • 10b = 1.7, a • 60b = 105.7 Dividing, 6b = 12. log y a= 1 e. 14c: 0.0084…(90)2.3049… = 269.1302… lb. 14d: 0.0084…(90)2.3033… = 269.5358… lb. 13. a. y = 5 • 3x ⇒ log y = log (5 • 3x) = log 5 + (log 3)x b. y = ab x ⇒ log y = log (ab x) = log a + (log b)x Log-log and Semilog Graph Paper Explorations c. y = 2x 3 ⇒ log y = log (2x 3) = log 2 + 3 log x d. y = ab x ⇒ log y = log (ax b) = log a + b log x e. y = a + c logb x = a + c • 1.7 1.7 = = 0.0084…; y = 0.0084…x2.3049… 10b 102.3049… d. y = 1.3499…(1.0820…)x has r = 0.9670… y = 0.0084…x2.3033… has r = 0.9999… Power regression equation is very close to the equation in part b and has a very good r-value. log x 1 105.7 105.7 ⇒ b = log6 = 2.3049…; 1.7 1.7 log x c =a+ • log x log b log b Log-log and Semilog Graph Paper Problem 1. LinReg: f (x) = 22.22220…x D 99.6007…, r = 0.8956… ExpReg: f (x) = 1.1099… • 1.5492…x, r = 0.9999… PwrReg: f (x) = 0.1561…x2.7734…, r = 0.9799… The exponential function fits the best. 14. a. Add-multiply semilog: y 1000 100 10 x 1 0 5 38 / Log-log and Semilog Graph Paper Solutions 10 Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Foerster_IRB2_032-040 9/24/03 8:28 PM Page 39 7. LinReg: h(x) = 1.3x + 4, r = 1 ExpReg: h(x) = 5.7733… • 1.0995…x, r = 0.9697… PwrReg: h(x) = 5.0032… • x 0.5517…, r = 0.9894… The linear function fits the best. 2. Points are in a line. f (x) 1000 8. The points are in a line. h(x) 30 100 25 20 10 15 x 1 5 10 15 10 0 3. f (0) = 1.1099…; f (15) = 788.7983… Points lie on the line. 4. LinReg: g(x) = D1.4425…x + 118.4618…, r = D0.5931… ExpReg: g(x) = 93.6080… • 0.9597…x, r = D0.9194… PwrReg: g(x) = 598.1917…xD1.1978…, r = D0.9999… The power function fits the best. 5 x 5. The points are in a line. 5 g(x) 10 15 20 9. h(0)=4, h(50)=69 The points lie on the line. 1000 10. Linear: Arithmetic graph paper Exponential: Add-multiply semilog graph paper Power: Log-log graph paper 11. Exponential function (the x is in the exponent). 12. log y = log(13 R 7x) = log 13 + x log 7 100 13., 14., 15. 10 x 1 10 100 1 6. g(1) = 598.1917…, g(50) = 5.5163… Points lie on the line. y-values Distance Ratio log 1 to 2 11.4 0.30 0.30 1 to 3 18.1 0.48 0.48 1 to 4 22.8 0.60 0.60 1 to 5 26.5 0.70 0.70 1 to 6 29.5 0.78 0.78 1 to 7 32.5 0.86 0.85 1 to 8 34.2 0.90 0.90 1 to 9 36.2 0.96 0.95 1 to 10 37.9 1.00 1.00 16. The distances on the vertical axis represent the common logarithm of the function’s value. 17. Answers will vary. Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press Log-log and Semilog Graph Paper Solutions / 39 Foerster_IRB2_032-040 9/24/03 8:28 PM Page 40 Mathematical Models and Log Graphs Problem 1. 7. g(x) 10000 f (x) 1000 1000 100 10 100 x 1 10 5 0 15 x 10 10 1 2. +2 +2 +2 +2 x f (x) 3 52.7 5 28.8 7 15.7 9 8.6 11 8. x (ft) × 0.546 ×2 × 0.546 × 0.546 ×2 × 0.546 4.7 Adding 2 to x multiplies f (x) by 0.546. 3. f (x) = 130.3922… • 0.7392…x, r = 0.9999… 4. f (0) = 130.3922… (See square point on graph in Problem 1.) The coffee was about 130.4− + 70−, or about 200.4−. 5. y = 2.1152… D 0.1312x, r = 0.9999… 100 g(x) 5 75 10 600 15 2025 20 4800 ×8 ×8 9. g(x) = 0.6x3, r = 1 The power function fits exactly because the correlation coefficient is exactly 1. 10. g(8) = 0.6(8)3 = 307.2 (See square point on graph in Problem 7.) 11. g(100) = 0.6 • 1003 = 600000 lbs = 3000 tons 3 12. g(x) = 3x + log 0.6, r = 1 The correlation coefficient is exactly 1, so the linear function fits exactly. 2 13. 6. log f(x) log f (x) 4 1 x 5 10 15 3 2 x 1 2 14. Answers will vary. 40 / Log-log and Semilog Graph Paper Solutions Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2 ©2004 Key Curriculum Press