UNIT TWO EXPONENTS AND LOGARITHMS MATH 621B 20 HOURS

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1 UNIT TWO EXPONENTS AND LOGARITHMS MATH 61B 0 HOURS Revised Apr 9, 0 9

2 SCO: By the end of grade 1, students will be epected to: B30 understand and use zero, negative and fractional eponents Elaborations - Instructional Strategies/Suggestions Introduction In this unit students will be challenged to discover that eponents and logarithms are inetricably interconnected. Many times students do not see the purpose of logarithms. Upon completion of this unit the applications of logarithms should be clear to students. As a review of eponents invite student groups to do the Mental Math eercises listed in the Suggested Resources column. One application of eponential equations is the compound interest formula A = P( 1 + i) n where A = the final amount P = the initial amount i = annual interest rate/ # times calculated per year n = total number of times interest is calculated Challenge student groups to do the Warm Up eercises listed in the Suggested Resources column. Later in the unit students will encounter problems where the unknown is in the eponent and logarithms will be used to solve for the unknown. 30

3 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Introduction Evaluate: a) 8 3 Introduction Math Power 1 Mental Math p.71 #19-35 odd Warm Up p.71 #1-5 odd 5 b) 814 Epress as a power of 3: 3 a) 7 4 b) 81 F HG I K J F 5 1 HG I 7K J 1 c) d) 9 The annual interest rate is 1%. Find the value of i if interest is calculated(compounded): a) annually b) semi-annually c) quarterly d) monthly How many times as a total is interest calculated if: a) it is compounded quarterly for 8 years b) it is compounded semi-annually for 5 years c) it is compounded monthly for 3 years If the rate of interest is 8% per annum, find the amount owing on a 6 year loan of $9000 if interest is compounded quarterly? 31

4 SCO: By the end of grade 1, students will be epected to: C68 describe general properties of linear, quadratic, eponential and logarithmic functions and visualize their graphs C69 eplore rates of growth and decay in the contet of functions Elaborations - Instructional Strategies/Suggestions Eponential Functions (.) An ecellent beginning to this section would be the Eploring Eponential Functions investigation on p.76. This is an etension of unit 6 in Math 51B (Developing a Function Toolkit). See the Suggested Resources column for the list of eercises. Challenge students to do the Eplore and Inquire on p.77. Invite students to read and discuss p The general form of an eponential function f ( ) = Ab or on the TI-83 y = ab where a and b are constants and b > 0. If b > 0 it is an eponential growth problem If 0 < b < 1 it is an eponential decay problem. E:1 Mary invests $5000 at 5% per annum compounded annually. Write an eponential function to model this situation from A = P( 1 + i) n Find when the investment has doubled. Solution n y = f ( ) = Use Trace to find the solution. E. John buys a new vehicle for $8,000. It depreciates by 15% each year. Write an eponential function that models this situation. What is the vehicle worth in 7 years time? Solution n y = Use Trace to find the solution. Note: Many of these graphs are drawn as continuous but the interest problem for instance is actually a discrete graph where the interest is calculated only once a year. The last section in this unit deals with continuous eponential functions. 3

5 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Eponential Functions (.) Journal In the epression y = b what effect does changing the b value have? What effect does adding a constant term to the epression have (e. y = b + c ). Group Activity The world s population is growing eponentially. In 1970 it was about 3.6 billion. If the population increased at % per year since then, what will the population be in 010? About 0.84 ha of land is needed to sustain one person. If there are about 5 billion hectares of arable land on Earth, what is the maimum sustainable population? In what year will this population be reached? (Hint:Use a TI-83 to solve this problem, don t use guess and check.) Eponential Functions Math Power 1 eploration p.76 #1-6 p.81 #13-15, odd, 3,33 omit 9 Problem Solving Strategies p.85 #1,3,7,11,13,14 Solution P = n The maimum sustainable population is = 30.3 b. Therefore the maimum sustainable population will be reached about 107 year after 1970 or in 077. Journal What other factors can be brought into play to avoid this future crisis? 33

6 SCO: By the end of grade 1, students will be epected to: B solve eponential equations both with and without technology Elaborations - Instructional Strategies/Suggestions Solving Eponential Equations (.4) In this section students will meet eponential equations that can be solved by getting terms to a common base. This will lead to problems where this method is not possible and logarithms must be used. Challenge student groups to do the eplore problem and answer the inquire questions on p.86. Invite student groups to read and discuss e. 1-3 on p Two of the applications are doubling periods for bacteria and viruses and half life for radioactive elements. Doubling formula Half-life formula N t = N d 0( ) N N t h F1 = H G I K J 0 where N 0 = number of items at time zero N = number at time t h = time for substance to decrease to half the original amount d = time for population to double 34

7 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Solving Eponential Equations (.4) Research Write a paper on viruses, their discovery, their role in diseases and include images of the shapes of some viruses. Solve: a) 7 = 9 3 Solving Eponential Equations Math Power 1 p.89 #1-5 odd,6,7 p.91 Green #1 1-7 Green # b) 16 = 3 Note: Use the TI-83 for the following, not guess and check. The trout population in a lake has been doubling every five years. There were 00 trout in the initial count. a) Write an equation that describes the growth using time t in years since the initial count. b) How many trout will there be in 1 years? c) How many trout were there 3 years ago? The half-life of tritium is about 1 years. A research scientist has a 00 g sample of tritium. a) Write an equation that describes the decay. b) How much tritium will be left after 9 years? after 50 years? 35

8 SCO: By the end of grade 1, students will be epected to: B33 discover, eplain and use eponential and logarithmic relationships C60 describe quadratic, eponential and logarithmic relationships and translate among the various representations Elaborations - Instructional Strategies/Suggestions Eploring Inverses of Functions Challenge students to graph y =, determine the equation of its inverse and remember from Unit 1 that they are reflections of each other across the y = line. Invite students to complete the following tables of values and graph on the same coordinate grid. y = y = C71 demonstrate an understanding that a logarithmic function is the inverse of an eponential function The eercises in the Inverse Eploration are a good lead-in to the net section on logarithms. 36

9 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Eploring Inverses of Functions /Technology Complete the following tables of values and sketch the graphs on the same ais. y = y = Eploring Inverses of Functions Math Power 1 p.93 Green 1 #1 a,b a-c Green # 1, What is the equation of the reflection line? Write to eplain why these are inverses of each other. Technology Use the TI-83 to graph y = 3. Then graph its inverse using the following instructions. Check your work with a Reflect- View. Press nd Draw 8:Draw Inv vars < yvars 1:Function 1:Y 1 and press enter. Can you epress in words how this inverse is related to the original function? Reflect-View Use a Reflect-View to sketch the inverse of the following eponential function y =

10 SCO: By the end of grade 1, students will be epected to: B3 understand the relationship that eists between the arithmetic operations and the operations on eponents or logarithms C63 apply logarithms and solve equations Elaborations - Instructional Strategies/Suggestions Logarithms (.5) In the previous section, problems were of an introductory nature such that both sides of the equation could be reduced to a common base. E = = = 1 16 = 3 The net level problem is one such as: = 1 Method 1 Using guess and check, estimate to be between 3 and 4. Through further guessing the estimate can be improved upon. Method Graph y = on the TI-83. To find the value that will yield a y value of 1 graph y = 1 and determine the intersection point of these two equations. For the function y =, the inverse is = y. This inverse cannot be solved for y with any operations known to date. We need a convenient way of writing y is the eponent for base that gives an answer of. Mathematicians created a new operation and called it the logarithm operation. Therefore the inverse of an eponential function can be called a logarithmic function.. Invite students to do the Eplore and Inquire eercise on p.94. A Logarithm is an eponent. log b n = b = n where b> 0, b 1 Since eponential growth is a rapid rate of growth an logarithmic scale is used to make the numbers more manageable. Eamples are the Richter scale, the Decibel scale and the ph scale. Note to Teachers: Unless a base is specified, when log is written it is assumed to be a base 10 or common logarithm. Calculators(including the TI-83) can work directly with only common and natural logarithms. 38

11 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Logarithms (.5) Research Write a short paper on the life of John Napier and his contributions to mathematics. Epress in logarithmic form: a) 6 = 36 1 b) 5 = 5 Logarithms Math Power 1 p.98 #1-39,46,47 See Graphs of Logarithmic Functions Investigation at the end of the unit. Epress in eponential form: a) log 3 81 = 4 b) log = Evaluate: a) log 5 15 b) log 64 = Research Use the internet to research the Richter, Decibel and ph scales. 39

12 SCO: By the end of grade 1, students will be epected to: C7 demonstrate an understanding of the properties of logarithms and apply them Elaborations - Instructional Strategies/Suggestions Laws of Logarithms (.6) Invite student groups to read p where 4 logarithm laws and eamples of their use are shown. The eamples and problems in this section are a lead-in to being able to handle logarithmic equations with multiple terms which will be encountered in the net section. Eponent Law Logarithm Law y + y Product law b b = b log y = log + log y b b b Quotient law b b y = b y log = log log y b b b y Power law ( b ) y y = b log = n log b n b y y 1 Root law b = b log n b = log n 1 b Two other helpful rules are: a 1 = a log a a = 1 Change of base: log b a = 1 log a b 40

13 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Laws of Logarithms (.6) Epress as a single logarithm and evaluate: a) log 80 log 10 b) log 1 log 4 3 c) log Laws of Logarithms Math Power 1 p.106 #1-41 odd Journal Eplain how the eponent laws and logarithm laws are equivalent. Research Write a short paper on the life and contributions of Henry Briggs. Activity The period of a pendulum can be given by the formula: 1 log T = log π + (log l log g). Epress the equation in eponential form. 41

14 SCO: By the end of grade 1, students will be epected to: C63 apply logarithms and solve equations C7 demonstrate an understanding of the properties of logarithms and apply them Elaborations - Instructional Strategies/Suggestions Logarithms and Equations Solving (.7) Logarithms are an important tool in: a)solving eponential equations with the unknown in the eponent. E. Solve: 3 = 7 log 3 = log 7 log 3 = log 7 log 7 = log 3 b) evaluating logarithmic epressions with base other than 10. E. Evaluate = log = 57 log 3 = log 57 log 3 = log 57 log 57 = log 3 c) solving logarithmic equations/scientific formulae E. Solve for : log ( + 3) + log ( ) = =!4, 3 can t take log of a negative number so!4 Students should read and discuss E. 6 on p.11 in order to appreciate the importance of restrictions in solving logarithmic equations. A worthwhile eercise might be for students to look at the Sound Intensity and Decibels Investigation on p An eplanation of simplified and the actual scientific formulae for decibels, richters and ph is given at the end of this unit. 4

15 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Logarithms and Equations Solving (.7) /Technology Solve for and state any restrictions: + 3 a) 4 = 7 b) log ( + 3) = log ( 4) 4 4 Group Activity A fossil was unearthed and tests done on it revealing that the 1 amount of Carbon 14 in it was of what would be 8 normally found in the atmosphere. If Carbon 14 has a half-life of 5760 years, how old is the fossil? Solution A t 1 8 log F1 h = A H G I K J 0 t t 5760 F = H GI 1 K J F1 HG I K J = 8 log F1 H G I K J t 5760 Logarithms and Equations Solving Math Power 1 p.113 #1-5 odd, 6-3,4-44, 40(a) -(f),46, 47,5,54 Applications of Logarithms at end of the unit. Math 1 (Addison Wesley) p.154 #,4 t log 8 = log t log = log t = 5760 t = 8460years 43

16 SCO: By the end of grade 1, students will be epected to: C69 eplore rates of growth and decay in the contet of functions Elaborations - Instructional Strategies/Suggestions Eponential Functions using base e (.8) Invite students to read and discuss p in Math Power 1. Any eponential function can be modelled using base e. The symbol e is used to denote the irrational number It s inverse is seen on scientific calculators as ln and is known as a natural logarithm (sometimes pronounced lawn). They are called natural logarithms because many times it is the natural choice for scientists and others to use. Note to Teachers: The main advantage of writing eponential functions with base e is that the constant in the eponent is the : growth rate (when the eponent is positive) decay rate (when the eponent is negative) E: The present population of PEI is 1.35 hundred thousand. The population increases at the rate of.% per year. Model the eponential function and find the predicted population 50 years from now. Discuss with the members in your group if this prediction is a reliable one. Method 1 Method P P = 135. ( 10. ) = 135. ( 10. ) 50 t P P = 135. e = 135. e.0t.0( 50).0 is the growth rate E: The intensity of sunlight below the water s surface is reduced by 4.6% for every metre below the surface. Model the eponential function and find the light intensity 0 metres down. Use I, I 0 and d in the model. Method 1 Method d.046d I = I 0 (. 954) I = I0e I 0 is the light intensity!.046 is the decay rate 44

17 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Eponential Functions using base e (.8) Technology Use a calculator to find to 3 places of decimals: a) ln 7. b) ln 71 c) e 0.4 d) e 48. /Technology Solve to 3 decimal places: 3 a) 483 = e b) ln 14. = 016. The number of bacteria in a culture was 000 at a certain time and increased at a rate of 5% per hour. How many bacteria will the culture contain 19 hours later? Eponential Functions using base e Math Power 1 p.119 #1-19 odd, 1-9,3,33 Archaeology p.11 Green 1 # Green #1 Group Activity A bacteria culture grows continuously according to the formula N = 7000e 0.17t, where t = hours elapsed. In how many hours will the culture contain 31,000 bacteria? Presentation Atmospheric pressure decreases according to the formula a P = 100e 0.139, where a = altitude in km, P = pressure in Kpa (kilopascals). What is the altitude where the pressure is 60 Kpa.? 45

18 Graphs of Logarithmic Functions A general form for a logarithmic function is: y = log( a + b) + c. On the TI-83, graph the following using this window: 1. a) y = log b) y = log( 05. ) c) y = log( ) d) What effect does a have on the graph of the basic logarithmic function, y = log?. a) y = log b) y = log( + 1) c) y = log( 1) d) What effect does b have on the graph of the basic logarithmic function, y = log? 3. a) y = log b) y = log + 1 c) y = log 1 d) What effect does c have on the graph of the basic logarithmic function, y = log? 46

19 1) Richter Scale # times stronger = 10 I )I is the difference between the Richter magnitudes of the quakes ) Decibel Scale Originally the unit for loudness was the Bel named for Aleander Graham Bell. This unit was too large for convenient use so the Bel was subdivided into 10 smaller units called decibels (in the metric system the prefies are mm, cm, dm (decimetres), m...). L # times louder = )L is the difference between the decibel (db) readings of the sounds. 3) ph Scale # times more acidic = 10 ph )ph is the difference in ph readings between the solutions. Note to Teachers: A formula for the decibel scale was given on p.14. It has been changed to the one shown above. If the formula in the tet was used and numbers from the Decibel Scale substituted into it, the math doesn t work. The above formulae will give a relative strength between situations( earthquakes, sounds, solutions). On the last page of this unit is an eplanation of how the scientific Decibel formula works. This is for teacher information and may be more applicable to a Physics class. It is included for teacher information and may be everything you wanted to know about Decibels but were afraid to ask. 47

20 The Decibel Scale The formula for intensity of sound can be written in eponential or logarithmic form as shown below: I I 0 db = db = 10 log I I 0 E: Two sounds have loudness levels of 40 db and 80 db. How many times as loud is the 80 db sound as the 40 db sound? If a student uses the formula I I 0 = db and substitutes the difference in loudness of 40 db into the 80 eponent of the formula = the math doesn t work. The numbers for I and I 0 should be in 40 the units of Watts/m rather than decibels. See the table below. Intensity (W/m ) Decibel Level (db) ! ! ! ! !10 0 If we use the actual values in the intensity column 10!1 0 we see that the formula does actually work. I I = 10 = db = = Applications of Logarithms

21 Logarithms allow us to work with very large or very small numbers in a more manageable way. E. 1: Eponential curve fitting using logarithms The graph of y = increases so rapidly that very little of it is visible even if the y-ais is scaled up to 100,000. Complete the table of values below: y= The graph is : 49

22 If we change the equation to y = log complete the table of values below: y= log Now the graph looks like this: One of the advantages of having a graph that is a straight line is that it can be analysed using grade 10 math knowledge. Since this graph has a y-intercept of 0" then b = 0 in the equation y = m + b. Taking ordered pairs from the above table of values and calculating the slope gets a result of m = Therefore y = m + b = To appreciate what this means, look at the original equation. y = and take the log of both sides log y = log = log comparing y = m and log y = log we see that m = log = To generalize; if we have a growth function y = Ab k we can plot log y against (rather than y 50

23 against ) and use our knowledge of straight lines to find the equation for the line of best fit. In other words, we can use this technique to fit an eponential curve to data because you can do all the fitting with the straight line and then convert everything back to eponential form. E. Power Curve fitting using logarithms For a circle A = πr. Assume for the moment we don t know the formula but want to determine this relationship from the data table below. Radius() Area(y) This is table of values composed of Radius and y = log B r then its graph would not be linear. not an eponential curve because if we developed a If instead we complete the following table of values: 51

24 log Radius(log) log Area(logy) log 1 log 3.14 log log 1.6 log 3 log 8.3 log 4 log 50.3 log 5 log 78.5 log 6 log log 7 log log 8 log 01.1 log 9 log 54.5 log 10 log 314. If this is graphed then we would see a straight line graph. If we notice where the graph intersects the y- ais we find it is If we get the slope of the line we get a value of. To see what this means look at the original equation. A = πr log A = log( πr ) log A = log π + log r log A = log r + log π y = m + b We see that m = and b = log B or Therefore if we have data that we suspect is not eponential but instead is defined by a power function then we can use the method of graphing the logarithms of both the and y to find the eponent (slope of the line) to which the variable is raised and the coefficient of the variable ( 10 raised to the power of the y-intercept. In our eample above = B. A A A b = 10 r = r = π r m 5