Functions Modeling Change A Preparation for Calculus Third Edition

Transcription

1 Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1

2 Chapter 4: Logarithmic Functions Section 4.1 Logarithms and Their Properties

3 Suppose that a population grows according to the formula P = 10 t, where P is the colony size at time t, in hours. When will the population be 2500? We want to solve which equation for t? Page 152 3

4 Suppose that a population grows according to the formula P = 10 t, where P is the colony size at time t, in hours. When will the population be 2500? We want to solve which equation for t? 10 t 2500 Page 152 4

5 Let's solve graphically via our calculator. 10 t 2500 Page 152 5

6 Let's solve graphically via our calculator. y 10 x Window Value Xmin -1 Xmax 4 Xscl 1 Ymin 0 Ymax 3000 Yscl 300 Page 152 6

7 Let's solve graphically via our calculator. y 10 x 3,000 2,000 1,000 Page x 7

8 But we really want What is the next step? 3,000 t ,000 1,000 Page x 8

9 But we really want What is the next step? 3,000 2,000 t Add y 2 = ,000 Page x 9

10 But we really want t ,000 2,000 1,000 Page x 10

11 But we really want t ,000 Next? 2,000 1,000 Page x 11

12 But we really want t ,000 2nd Trace Intersect 2,000 1,000 Page x 12

13 But we really want t ,000 x= , y=2500 2,000 1,000 Page x 13

14 The central question is: Is there another way to solve this problem without graphics on our calculator? 3, t ,000 1,000 Page x 14

15 To answer this question, we define the common logarithm function, or simply the log function, written: log 10 x, or log x, as follows: Page

16 If x is a positive number, log x is the exponent of 10 that gives x. In other words, if y log x then 10 y x Page 152 Blue Box 16

17 To what power must 10 be raised to = 100? Page

18 To what power must 10 be raised to = 100? 2 "Log100 is the exponent of 10 that gives 100." 2 log100 2 means: if y log x then 10 x y if 2 log100 then Page

19 To what power must 10 be raised to = 10? Page

20 To what power must 10 be raised to = 10? 1 "Log10 is the exponent of 10 that gives 10." 1 log10 1 means: if y log x then 10 x y if 1 log10 then Page

21 To what power must 10 be raised to = 1? Page

22 To what power must 10 be raised to = 1? 0 "Log1 is the exponent of 10 that gives 1." 0 log1 0 means: 10 1 if y log x then 10 x y if 0 log1 then Page

23 To what power must 10 be raised to =.1? Page

24 To what power must 10 be raised to =.1? -1 "Log.1 is the exponent of 10 that gives.1." 1 log.1 1 means: 10.1 if y log x then 10 x y if 1 log.1 then Page

25 To what power must 10 be raised to =.01? Page

26 To what power must 10 be raised to =.01? -2 "Log.01 is the exponent of 10 that gives.01." 2 log.01 2 means: if y log x then 10 x y if -2 log.01 then Page

27 Back to our problem: 10 t 2500 How can we use logarithms to solve? Page

28 To what power must 10 be raised to = 2500? Page

29 To what power must 10 be raised to = 2500?? "Log2500 is the exponent of 10 that gives?" Page

30 To what power must 10 be raised to = 2500? "Log2500 is the exponent of 10 that gives " log means: y if y log x then 10 x if log 2500 then Page

31 Rewrite the following statements using exponents instead of logs. (a) log 100 = 2 (b) log 0.01 = 2 (c) log 30 = Page 152 Example #1 31

32 Rewrite the following statements using exponents instead of logs. (a) log 100 = 2 (b) log 0.01 = 2 (c) log 30 = if y log x then 10 y x Page 152 Example #1 32

33 Rewrite the following statements using exponents instead of logs. (a) log 100 = 2 (b) log 0.01 = 2 (c) log 30 = (a) 2 = log = 100. Page 152 Example #1 33

34 Rewrite the following statements using exponents instead of logs. (a) log 100 = 2 (b) log 0.01 = 2 (c) log 30 = (a) 2 = log = 100. (b) 2 = log = Page 152 Example #1 34

35 Rewrite the following statements using exponents instead of logs. (a) log 100 = 2 (b) log 0.01 = 2 (c) log 30 = (a) 2 = log = 100. (b) 2 = log = (c) = log = 30. Page 152 Example #1 35

36 Rewrite the following statements using logs instead of exponents. (a) 10 5 = 100,000 (b) 10 4 = (c) = if 10 y x then y log x Page 152 Example #2 36

37 Rewrite the following statements using logs instead of exponents. (a) 10 5 = 100,000 (b) 10 4 = (c) = (a) 10 5 = 100,000 log 100,000 = 5. Page 152 Example #2 37

38 Rewrite the following statements using logs instead of exponents. (a) 10 5 = 100,000 (b) 10 4 = (c) = (a) 10 5 = 100,000 log 100,000 = 5. (b) 10 4 = log = 4. Page 152 Example #2 38

39 Rewrite the following statements using logs instead of exponents. (a) 10 5 = 100,000 (b) 10 4 = (c) = (a) 10 5 = 100,000 log 100,000 = 5. (b) 10 4 = log = 4. (c) = log = 0.8. Page 152 Example #2 39

40 Logarithms are just exponents!!! Page

41 Without a calculator, evaluate the following, if possible: (a) log 1 (b) log 10 (c) log 1,000,000 (d) log (e) log 10 (f) log( 100) Page 153 Example #3 41

42 (a) log 1 = 0, since 10 0 = 1.? 10 1 Page 153 Example #3 42

43 (b) log 10 = 1, since 10 1 = 10.? Page 153 Example #3 43

44 (c) Since 1,000,000 = 10 6, the exponent of 10 that gives 1,000,000 is 6. Thus, log 1,000,000 = 6.? 10 1,000,000 Page 153 Example #3 44

45 (d) = 10 3, the exponent of 10 that gives is 3. Thus, log = 3.? Page 153 Example #3 45

46 (e) Since , the exponent of 10 that gives is -1/2. Thus log( ) = -1/2.? Page 153 Example #3 46

47 (f) Since 10 to any power is positive, 100 cannot be written as a power of 10.? Page 153 Example #3 47

48 (f) Since 10 to any power is positive, 100 cannot be written as a power of 10. Thus, log( 100) is undefined.? Page 153 Example #3 48

49 Logarithmic and Exponential Functions are Inverses For any N, log 10 N N and for N > 0, 10 log N N Page 153 Blue Box 49

50 Evaluate without a calculator: (a) log ( ) (b) 10log 2.7 (c) 10log(x + 3) Page 153 Example #4 50

51 Using log(10 N ) = N and 10 logn = N, we have: (a) log ( ) = 8.5 (b) 10 log 2.7 = 2.7 (c) 10 log(x + 3) = x + 3 A calculator will confirm (a) & (b). Page 153 Example #4 51

52 Properties of the Common Logarithm By definition, y = logx means 10 y = x. In particular, log 1 = 0 & log 10 = 1. The functions 10 x and logx are inverses, so x they undo each other: log 10 x for all x, log x 10 x for x 0 Page 154 Blue Box 52

53 Properties of the Common Logarithm For a and b both positive and any value of t, log ab log a logb a log log a log b b log t b t logb Page 154 Blue Box 53

54 Properties of the Common Logarithm log ab log a log b, note: log( a b) log a log b a log log a log b, note: log( a b) log a log b b a log a and note: log b logb t log b t log b Page 154 Blue Box 54

55 Solve (100)(2 t )=337,000,000 for t Page 154 Example #5 55

56 Solve (100)(2 t )=337,000,000 for t t (100)(2 ) t t log 2 log Page 154 Example #5 t log 2 log t log log

57 The Natural Logarithm When e is used as the base for exponential functions, computations are easier with the use of another logarithm function, called log base e. The log base e is used so frequently that it has its own notation: ln x, read as the natural log of x. Page

58 The Natural Logarithm Definition: ln x is the power of e that gives x or ln x = y means e y = x y is called the natural logarithm of x Page

59 The Natural Logarithm Just as the functions 10 x and logx are inverses, so are e x and ln x. The function ln x has similar properties to the common log function: Page

60 Properties of the Natural Logarithm By definition, y = ln x means x = e y. In particular, ln 1 = 0 & ln e = 1. The functions e x and ln x are inverses, so they x undo each other: ln e x for all x, ln x e x x for 0 Page 155 Blue Box 60

61 Properties of the Natural Logarithm For a and b both positive and any value of t, ln ab ln a ln b a ln ln a ln b b ln t b t ln b Page 155 Blue Box 61

62 Properties of the Natural Logarithm ln ab ln a ln b, note: ln( a b) ln a ln b a ln ln a ln b, note: ln( a b) ln a ln b b a ln a and note: ln b ln b t ln b t ln b Page 155 Blue Box 62

63 Solve for x: (a) 5e 2x = 50 (b) 3 x = 100. Page

64 Solve for x: (a) 5e 2x = 50 Page

65 Page 155 Solve for x: (a) 5e 2x = 50 ln 2x 5e 50 e e 2x 2x 10 ln(10) 2x ln 10 x ln

66 Solve for x: (b) 3 x = 100. Page

67 Solve for x: (b) 3 x = 100. ln 3 x xln 3 x x x ln(100) ln(100) ln(100) ln(3) Page

68 Note: log( a b) log a logb log( a b) log a logb log( ab) (log a)(log b) log a b log a logb log 1 1 a log a Page 156 Blue Box 68

69 Note: 2 log5x 2log(5 x) (since exponent only applies to the x) 2 2 log5x log(5) log( x ) log(5) 2log( x) Page 156 Blue Box 69

70 Note: ln( a b) ln a ln b ln( a b) ln a ln b ln( ab) (ln a)(ln b) ln a b ln ln a b ln 1 1 a ln a Page 156 Blue Box 70

71 End of Section