Supplementary Lesson: Log-log and Semilog Graph Paper

Transcription

1 Supplementar Lesson: Log-log and Semilog Graph Paper Chapter 7 looks at some elementar functions of algebra, including linear, quadratic, power, eponential, and logarithmic. The following supplementar section on log-log and semilog graph paper is intended for use at the conclusion of this chapter. This section provides teaching materials, eplorations, teacher notes, and solutions. You can use this lesson in our regular teaching plan or assign it for individual projects, group investigations, or etra credit.

2 Instructor s Commentar Objective Given an eponential, power, or logarithmic function, plot its graph on semilog or log-log graph paper. Class Time 2 das Homework Assignment Da : Q Q, Problems, 3,, 7 Da 2: Problems 9,, 3, 4 Important Terms and Concepts Semilog graph paper Log-log graph paper Lesson Notes Eponential functions can be ver hard to read for points with large -values and for -values near the horizontal asmptote. One wa of showing the function more clearl is to change the -scale to log. This makes the -values smaller for large -values and negative for positive values of near zero. Students ma have seen such graphs before. For instance, graphs displaing 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 eponential function = (.6 ) is linear when graphed on semilog paper. Therefore, it follows that if the points plotted on the eponential function were re-epressed as (, log ), then these re-epressed points would be linear on arithmetic graph paper. Similarl, power functions are linear on log-log paper, and a reepression of the power data as (log, log ) would be linear on arithmetic graph paper. In Eample, 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 multipl multipl relation in the and variables. Students will revisit parametric equations and see how to use them to graph the log log relationship. Eample 2 demonstrates how to use semilog paper to graph a multipl add relationship, which models a logarithmic function. 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 ma get an error message when the are doing a regression on their grapher. When the grapher does eponential regression, it re-epresses the data using (, log ), which is wh the grapher is unable to do eponential regression on a data set that includes as one of the -values. Likewise, power regression re-epresses the data using (log, log ), so and must be positive numbers. Eploration Notes In Mathematical Models and Log Graphs, students will plot points on semilog and log-log graph paper and make connections to the add multipl (eponential) and multipl multipl (power) properties. Allow about 2 minutes to complete. It ma not be obvious which propert a set of data ehibits, so Log-log and Semilog Graph Paper gives students practice in finding which regression best fits the data. The results are then verified b using the appropriate graph paper to show the linear relationship. This eploration should take about 2 minutes to complete. Problem Notes Problems 8 reinforce the skills and concepts in the eamples and eplorations. Problems 9 4 demonstrate that the log versus is linear for eponential functions and that the log versus log is linear for power functions. Precalculus with Trigonometr: Instructor s Resource Book, Volume 2 Log-log and Semilog Graph Paper Instructor s Commentar / 2

3 Log-log and Semilog Graph Paper Student Lesson In man real-world situations, the -values of a function span a wide range. For instance, suppose the number of bacteria in a culture at the end of an one hour has dropped to 6% of what it was at the beginning of the hour. If there were initiall million bacteria present, then the number of millions of bacteria,, is given b the eponential function =.6 where is time in hours. Figure A Figure B Figure A shows the graph of this function for the first hours. It is hard to read the graph for between and, even though the aes have a large vertical dimension. In Figure B the vertical scale has been compressed for larger -values and epanded for smaller -values. The spaces represent the logarithms of the -values. You ma alread have seen such semilog graph paper in the eplorations. On this graph it is just as eas to read for larger -values. For eponential 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 -direction also represent logarithms, of the -values. OBJECTIVE Given an eponential, 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 Trigonometr: Instructor s Resource Book, Volume 2

4 Straight-line Semilog and Log-log Graphs To see wh an eponential function has a straight-line graph on semilog paper, it is sufficient to show that log varies linearl with. To do this, take the log of both sides of the eponential function equation. =.6 log = log (.6 ) Write log in front of both sides. log = log + log.6 log = 3 D.87 N log varies linearl with, Q.E.D. The log of a product equals the sum of the logs. log = log + log.6 The log of a power equals the eponent times the log of the base. B calculator Form: = a + b The graph in Figure B is a straight line with a negative slope. In Problem 3 of this lesson ou ll prove this propert in general for eponential functions. In that problem ou ll also prove that the graph of a power function is a straight line on log-log graph paper. U EXAMPLE Snake Skin Problem: Snakes shed their skins periodicall. The area of the skin depends on the length of the snake. Suppose ou measured the areas in this table. Length (cm) Area (cm 2 ) a. Plot the data on log-log graph paper. What do ou notice about the points? b. Show b regression that a power function fits these data. Write its particular equation. c. Predict the skin area of an anaconda snake 7 meters long. Surprising? d. Use our equation from part b to plot on our grapher the log of area as a function of log of length. What do ou notice about the graph? e. Prove algebraicall that log of area is a linear function of log of length. What do ou notice about the slope of this graph? Solution a. Let = length in cm and let = area in cm 2. Figure C shows the graph of area as a function of length on log-log paper. The points lie in a straight line. Figure C Precalculus with Trigonometr: Instructor s Resource Book, Volume 2 Log-log and Semilog Graph Paper Student Lesson / 27

5 b. B power regression, r =. So a power function fits the data. The particular equation is =.9 2. c. 7 meters is 7 cm, so =.9(7 2 ) = 44, cm 2. Surprising! d. Put our grapher in parametric mode. Then enter the equations like this: = log t = log (.9 t 2 ) The graph is shown in Figure D. It is a straight line. 3 log 2 log 2 Figure D e. log = log (.9 2 ) Take the log of both sides. log = log log log = D log N log is a linear function of log, Q.E.D. log of a product and log of a power B calculator Form: log = a(log ) + b The slope of the linear graph equals the eponent of. You can confirm this fact geometricall b picking two points on the graph, measuring the rise run and the rise with a ruler, and observing that run =2. Figure E shows how ou can do this Ruler Figure E V 28 / Log-log and Semilog Graph Paper Student Lesson Precalculus with Trigonometr: Instructor s Resource Book, Volume 2

6 U EXAMPLE 2 a. B regression analsis, show numericall that a logarithmic function fits these - and -values better than a linear, eponential, or power function b. Write the particular equation of the best-fitting logarithmic function. c. Plot the points on multipl-add semilog graph paper. Show that the points lie on a line. d. Calculate if is 2. Show that this point lies on the graph in part c. e. Calculate if is. Solution a. Linear: r =.833 Logarithmic: r = Best fit! Eponential: r =.733 Power: r =.9736 b. Equation is = ln. Paste it into the = menu (without rounding). c. Figure F shows that the points lie in a straight line. You can get multipladd semilog graph paper b rotating regular add-multipl paper 9 clockwise. Regression line fits closel. (2, 6...) is on the line. Figure F d. If = 2, then = ln 2 = 6. The point (2, 6. ) is on the regression line in Figure F. e. If =, then = ln Use the solver to find numericall. V Precalculus with Trigonometr: Instructor s Resource Book, Volume 2 Log-log and Semilog Graph Paper Student Lesson / 29

7 DEFINITIONS: Kinds of Graph Paper Arithmetic graph paper ( ordinar graph paper) has linear scales on both aes. Log-log graph paper has logarithmic scales on both aes. Add-multipl semilog paper has a logarithmic scale on the vertical ais onl. Multipl-add semilog paper has a logarithmic scale on the horizontal ais onl. Problem Set Do These Quickl Q. log + log 7 = log? Q2. log 8 D log 3 = log? Q3. 2 log 7 = log? Q4. log =? Q. log =? Q6. Logarithmic functions have the? D? pattern. Q7. What pattern do the -values of a quadratic function follow for regularl spaced -values? Q8. Name the kind of sequence: 23, 3, 37, 44, Q9. 6 =? min Q. Find the area of a right triangle with one leg 3 cm and hpotenuse cm. For the eponential functions in Problems and 2, a. Calculate the -values for the given values of. b. Plot the points on add-multipl semilog graph paper. c. Show that the points lie on a straight line.. =. 2 ; =, 3,, 7, 9 2. = 8.6 ; = 2, 4, 6, 8, For the power functions in Problems 3 and 4, a. Calculate the -values for the given values of. 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 eponent in the equation. 3. = 7 D.3 ; =,,, 3, 4. =.8 ; =, 6,, 4, For the logarithmic functions in Problems and 6, a. Calculate the -values for the given values of. b. Plot the points on multipl-add semilog graph paper. c. Show that the points lie on a straight line.. = ln ; =, 4,, 2, 6. = D + 2 log ; = 3, 8, 2,, 7. Show that an eponential function graph is not straight on log-log paper b plotting the data from Problem on log-log paper. 8. Show that a power function graph is not straight on semilog paper b plotting the data from Problem 4 on semilog paper. For the eponential functions in Problems 9 and, plot on our grapher log as a function of. Use suitable - and -windows. Sketch the result. 9. = 3. = 2.8 For the power functions in Problems and 2, plot on our grapher log as a function of log. You ma use parametric mode as in Eample. Use suitable - and -windows. Sketch the result.. = 9 D2 2. = Proof Problems: a. Prove that for the eponential function = 3, log is a linear function of. b. Prove in general that for the eponential function = ab, log is a linear function of. 3 / Log-log and Semilog Graph Paper Student Lesson Precalculus with Trigonometr: Instructor s Resource Book, Volume 2

8 c. Prove that for the power function = 2 3, log is a linear function of log. d. Prove in general that for the power function = a b, log is a linear function of log. e. How can ou conclude that in a logarithmic function, is a linear function of log? 4. Height-Weight Historical Problem: Before there were calculators to do regression analsis efficientl, 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) a. Plot the data on add-multipl semilog paper. Plot it again on log-log paper. Which graph seems to be more nearl a straight line? b. Based on our answers to part a, what kind of function fits the data more closel, eponential or power? c. Find algebraicall 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 wa particular equations were found in the das before calculators had regression analsis built in.) d. Do eponential regression and power regression on the given data. Does the regression analsis confirm our conclusion and equation? How can ou tell? e. Predict the weight of a 9-in.-tall giant using our equation from part c and again using the regression equation in part d. How closel do the two answers agree? Precalculus with Trigonometr: Instructor s Resource Book, Volume 2 Log-log and Semilog Graph Paper Student Lesson / 3

9 Name: Group Members: Eploration: Log-log and Semilog Graph Paper Date: Objective: Given a table of data with irregularl spaced -values, find b 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 3, use this data. f() B regression, find the particular equation of the best-fitting linear, eponential, or power function. For Problems 4 6, use this data. g() B regression, find the particular equation of the best-fitting linear, eponential, or power function. 2. Plot the data on this semilog graph paper. What do ou notice about the points? f(). Plot the data on this log-log graph paper. What do ou notice about the points? g() 3. Calculate f () and f (). Plot these points on the graph. Do these points lie on the same straight line as the given data points? 6. Calculate g() and g(). 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 Eplorations Precalculus with Trigonometr: Instructor s Resource Book, Volume 2

10 Name: For Problems 7 9, use this data. Group Members: Eploration: Log-log and Semilog Graph Paper continued h() B regression, find the particular equation of the best-fitting linear, eponential, or power function. Date:. What kind of graph paper gives a straight-line graph for Linear functions? Eponential functions? Power functions?. Let = 3 7. What kind of function is this? How do ou tell? 8. Plot the data on this arithmetic graph paper. What do ou notice about the points? h() 2 9. Calculate h() and h(2). Plot these points on the graph. Plot a straight line through these points. How does the given data relate to the line? 2. Take the log of both sides of the equation in Problem. B appropriate use of the properties of logs, show that log varies linearl with. 3. From the semilog paper in Problem 2, measure to the nearest. mm the following distances on the vertical scale: -values Distance, mm Ratio log 2 (3, 4,...) to 2 to 3 to 4 to to 6 to 7 to 8 to 9 to 4. Make another column in the table above and record Distance from to 2 (3, 4, ) Distance from to Round the ratios to 2 decimal places.. Make another column in the table above and record log 2 (3, 4, ). That is, record log in the to row and so on. Round to 2 decimal places. 6. Based on the table above, what do the distances on the -scale of semilog paper represent? 7. What did ou learn as a result of doing this Eploration that ou did not know before? Precalculus with Trigonometr: Instructor s Resource Book, Volume 2 Log-log and Semilog Graph Paper Eplorations / 33

11 Name: Group Members: Eploration: Mathematical Models and Log Graphs Date: Objective: Plot graphs of eponential or power functions on semilog or log-log graph paper, respectivel. Coffee Cup Problem: You pour a cup of coffee. Three minutes after ou pour it, ou find that it is 2.7 F above room temperature. You record its temperature each 2 minutes thereafter, finding the following values: (min) f () ( F) Plot these points on this semilog graph paper. The scale on the -ais is logarithmic, with the distance from the -ais representing the logarithm of the -value. For between and, each space represents unit. For between and, each space represents units, and so forth. Connect the points. If an point does not lie on a straight line, go back and check our work. 3. Eponential functions have the add-multipl propert. B eponential regression find the particular equation of the best-fitting eponential function. Write the correlation coefficient. 4. Use our equation to find f (). Show that the answer agrees with the pattern of the points in Problem. If the room temperature was 7, how hot was the coffee when it was poured?. Put log f () in a third list in our grapher. Then do linear regression with and log f (). Write the result here. Write the correlation coefficient. f() 6. On our grapher, plot the linear function from Problem along with the points from the lists of and log f (). Use an -window of [, ] and a -window of [, 3]. Sketch the result here. What relationship do ou notice between this graph and the graph in Problem? 2. Show in the table that the points have, approimatel, the add-multipl propert. State the result verball. Adding 2 to multiplies f () b about?. 34 / Log-log and Semilog Graph Paper Eplorations Precalculus with Trigonometr: Instructor s Resource Book, Volume 2

12 Name: Group Members: Eploration: Mathematical Models and Log Graphs continued Date: Shark Problem: From great white sharks caught in the past, fishermen find the following weights and lengths. (ft) g() (lb) 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 our mistake. g(). 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.. From fossilized sharks teeth, naturalists think there were once great white sharks ft long. Based on our mathematical model, how heav would such a shark be? Surprising? 2. Put two more lists in our grapher for the shark data of Problem 7, one for log and one for log g(). Do linear regression for log g() as a function of log and write the equation. What evidence do ou have that a linear function fits these transformed values eactl? 3. On our grapher, plot the linear function along with the points from the lists of log and log g(). Use an -window of [, 2] and a -window of [, 4]. Sketch the result. How does the graph compare with the log-log graph in Problem 7? 8. Show in the table that the points have the multiplmultipl propert. State the result verball. Multipling b? multiplies g() b?. 4. What did ou learn as a result of doing this Eploration that ou did not know before? 9. Power functions have the multipl-multipl propert. Do power regression on the given points. Write the particular equation. What evidence do ou have that the fit is eact? Precalculus with Trigonometr: Instructor s Resource Book, Volume 2 Log-log and Semilog Graph Paper Eplorations / 3

13 Solutions Problem Set Q. log 3 Q2. log 6 Q3. log 49 Q4. Q. Q6. multipl-add Q7. Constant 2nd differences Q8. Arithmetic Q9. Q. 6 cm 2 b., c.. a b., c. 3. a b., c. 2. a d. Slope = D.3 36 / Log-log and Semilog Graph Paper Solutions Precalculus with Trigonometr: Instructor s Resource Book, Volume 2

14 4. a b., c. b., c. 7. d. Slope =.8. a b., c a. 3 D Precalculus with Trigonometr: Instructor s Resource Book, Volume 2 Log-log and Semilog Graph Paper Solutions / 37

15 9. log Log-log:. log. log log 2. log log 3. a. = 3 log = log ( 3 )=log + (log 3) b. = ab log = log (ab )=log a +(log b) c. =2 3 log = log (2 3 )=log log d. = ab log =log (a b )=log a + b log e. = a + c log b = a + c log log b = a + c log b log 4. a.add-multipl semilog: Log-log gives more nearl a straight line. b. Power. c. a b =.7, a 6 b =.7 Dividing, 6 b =.7 ;.7 b =log = a =.7 =.7 =.84 ; b = d. =.3499 (.82 ) has r =.967 = has r =.9999 Power regression equation is ver close to the equation in part b and has a ver good r-value. e. 4c:.84 (9) = lb. 4d:.84 (9) = lb. Log-log and Semilog Graph Paper Eplorations Log-log and Semilog Graph Paper Problem. LinReg: f () = D 99.67, r =.896 EpReg: f () = , r =.9999 PwrReg: f () = , r =.9799 The eponential function fits the best. 38 / Log-log and Semilog Graph Paper Solutions Precalculus with Trigonometr: Instructor s Resource Book, Volume 2

16 2. Points are in a line. f() 7. LinReg: h() =.3 +4, r = EpReg: h() = , r =.9697 PwrReg: h() =.32.7, r =.9894 The linear function fits the best. 8. The points are in a line. 3 h() f () =.99 ; f () = Points lie on the line. 4. LinReg: g() = D , r = D.93 EpReg: g() = , r = D.994 PwrReg: g() =98.97 D.978, r = D.9999 The power function fits the best.. The points are in a line. g() 9. h()=4, h()=69 The points lie on the line. 2. Linear: Arithmetic graph paper Eponential: Add-multipl semilog graph paper Power: Log-log graph paper. Eponential function (the is in the eponent). 2. log =log(3 R 7 )=log 3+ log 7 3., 4.,. 6. g() = 98.97, g() =.63 Points lie on the line. -values Distance Ratio log to to to to to to to to to The distances on the vertical ais represent the common logarithm of the function s value. 7. Answers will var. Precalculus with Trigonometr: Instructor s Resource Book, Volume 2 Log-log and Semilog Graph Paper Solutions / 39

17 Mathematical Models and Log Graphs Problem. f() 7. g() 2. Adding 2 to multiplies f () b f () = , r = f () = (See square point on graph in Problem.) The coffee was about , or about = 2.2 D.32, r = f() log f() g() =.6 3, r = The power function fits eactl because the correlation coefficient is eactl.. g(8) =.6(8) 3 =37.2 (See square point on graph in Problem 7.). g() =.6 3 =6 lbs = 3 tons 2. g() =3 +log.6, r = The correlation coefficient is eactl, so the linear function fits eactl. 3. (ft) log f() g() Answers will var. 4 / Log-log and Semilog Graph Paper Solutions Precalculus with Trigonometr: Instructor s Resource Book, Volume 2