1. Exponential and Logarithmic Functions By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: January 30, 2008

2. With your Graphing Calculator graph each of the following y = 2 x y = 3 x y = 5 x y = 1 x Determine what is happening when the base is changing in each of these graphs. Jeff Bivin -- LZHS

3. y = 2 x Jeff Bivin -- LZHS xy = 2 x y = 3 x -21/41/9 ½1/3 011 123 249 3827 y = 3 x

4. y = 2 x Jeff Bivin -- LZHS xy = 5 x y = 1 x -21/251 1/51 011 151 2251 31251 y = 3 x y = 5 x y = 1 x

5. y = 2 x Jeff Bivin -- LZHS y = 3 x y = 5 x y = 1 x Determine where each of the following would lie? y=10 x y=4 x y = (3/2) x y = 10 x y = 4 x y = (3/2) x

6. Jeff Bivin -- LZHS

7. f(x) = 2 x Jeff Bivin -- LZHS

8. f(x) = 2 x-3 Jeff Bivin -- LZHS

9. f(x) = 2 x+2 - 3 Jeff Bivin -- LZHS

10. f(x) = -(2) x-4 – 2 Jeff Bivin -- LZHS

11. Compound Interest You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly? A = Final amount = unknown P = Principal = $5000 r = rate of interest =.045 n = number of times compounded per year = 4 t = number of years compounded = 10

12. Compound Interest You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly? A = unknown P = $5000 r =.045 n = 4 t = 10

13. Compound Interest You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly? A = unknown P = $5000 r =.045 n = 4 t = 10 weekly? 52

15. With your Graphing Calculator graph each of the following y = 1 x y = (1/2) x y = (1/3) x Determine what is happening when the base is changing in each of these graphs. Jeff Bivin -- LZHS

16. y = 2 x Jeff Bivin -- LZHS x y = (½) x y = (1/3) x -249 23 011 1½1/3 2¼1/9 31/81/27 y = 3 x y = 5 x y = 1 x y = (1/3) x y = (½) x

17. f(x) = 2 -x = (1/2) x Jeff Bivin -- LZHS

18. f(x) = (½) x-3 - 2 = (2) -x+3 - 2 Jeff Bivin -- LZHS

20. A new Number We could use a spreadsheet to determine an approximation.spreadsheet

21. A new Number

22. y = 2 x Jeff Bivin -- LZHS xy = 2 x y = 3 x -2¼1/9 ½1/3 011 123 249 3827 y = 3 x y = e x Graph y = e x

23. Jeff Bivin -- LZHS y = e x y = e x+2 Graph: y = e x+2 x + 2 = 0 x = -2

24. Compound Interest-continuously You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded continuously? A = Final amount = unknown P = Principal = $5000 r = rate of interest =.045 t = number of years compounded = 10

25. Compound Interest-continuously You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded continuously? A = unknown P = $5000 r =.045 t = 10

26. Bacteria Growth You have 150 bacteria in a dish. It the constant of growth is 1.567 when t is measured in hours. How many bacteria will you have in 7 hours? y = Final amount = unknown n = initial amount = 150 k = constant of growth = 1.567 t = time = 7

27. Bacteria Growth You have 150 bacteria in a dish. It the constant of growth is 1.567 when t is measured in hours. How many bacteria will you have in 7 hours? y = unknown n = 150 k = 1.567 t = 7

29. y = 2 x Jeff Bivin -- LZHS xy -21/4 ½ 01 12 24 38 xy 1/4-2 ½ 10 21 42 83 x = 2 y

30. How do we solve this exponential equation for the variable y? y = 2 x x = 2 y Jeff Bivin -- LZHS

31. LOGARITHMS exponentiallogarithmic b > 0 A > 0 Jeff Bivin -- LZHS

32. exponentiallogarithmic Jeff Bivin -- LZHS

33. Evaluate Jeff Bivin -- LZHS

34. Evaluate Jeff Bivin -- LZHS

35. Evaluate Jeff Bivin -- LZHS

36. Evaluate Jeff Bivin -- LZHS

37. Evaluate Jeff Bivin -- LZHS

38. Evaluate Jeff Bivin -- LZHS

39. y = 2 x Jeff Bivin -- LZHS xy -21/4 ½ 01 12 24 38 xy 1/4-2 ½ 10 21 42 83 x = 2 y y=log 2 x

40. Jeff Bivin -- LZHS x y = log 2 x 1/4-2 ½ 10 21 42 83 y = log 2 x y = log 3 x y = log 5 x x = 2 y

41. Jeff Bivin -- LZHS x y = log ½ x 1/42 ½1 10 2 4-2 8-3 y = log ½ x x = (½) y

42. Solve for x log 2 (x+5) = 4 2 4 = x + 5 16 = x + 5 11 = x Jeff Bivin -- LZHS

43. Solve for x log x (32) = 5 x 5 = 32 x 5 = 2 5 x = 2 Jeff Bivin -- LZHS

44. Evaluate log 3 (25) = u 3 u = 25 3 u = 5 2 ?????? Jeff Bivin -- LZHS

46. Change of Base Formula Jeff Bivin -- LZHS

47. Evaluate log 3 (25) = 2.930 Jeff Bivin -- LZHS

48. Evaluate log 5 (568) = 3.941 Jeff Bivin -- LZHS

49. Properties of Logarithms Product Property Quotient Property Power Property Property of Equality Jeff Bivin -- LZHS

50. Product Property multiplicationaddition multiplicationaddition Jeff Bivin -- LZHS

51. Product Property Jeff Bivin -- LZHS

52. Quotient Property divisionsubtraction divisionsubtraction Jeff Bivin -- LZHS

53. Quotient Property Jeff Bivin -- LZHS

54. Power Property log b (m p ) log b (m p ) = plog b (m) p Jeff Bivin -- LZHS

55. Power Property Jeff Bivin -- LZHS

56. Property of Equality Jeff Bivin -- LZHS

57. Expand product property power property Jeff Bivin -- LZHS

58. Expand quotient property product property power property Jeff Bivin -- LZHS

59. Expand quotient property product property power property distributive property Jeff Bivin -- LZHS

60. Condense power property product property quotient property Jeff Bivin -- LZHS

61. Condense group / factor product property quotient property Power property Jeff Bivin -- LZHS

62. Condense re-organize group product property Power property quotient property Jeff Bivin -- LZHS

64. Solve for x Jeff Bivin -- LZHS Property of Equality

65. Solve for x checks! Jeff Bivin -- LZHS

66. Solve for x Jeff Bivin -- LZHS

67. Solve for n Jeff Bivin -- LZHS Condense left side Property of Equality

68. Solve for n checks! Jeff Bivin -- LZHS

69. Solve for n Jeff Bivin -- LZHS

70. Solve for x Jeff Bivin -- LZHS Condense left side Convert to exponential form

71. Solve for x checks! fails The argument must be positive Jeff Bivin -- LZHS

72. Solve for x Jeff Bivin -- LZHS

73. Solve for x Jeff Bivin -- LZHS Convert to exponential form

74. Solve for x checks! Jeff Bivin -- LZHS

75. Solve for x Jeff Bivin -- LZHS

76. Solve for x Jeff Bivin -- LZHS

77. Solve for x Jeff Bivin -- LZHS

78. Solve for x Jeff Bivin -- LZHS

79. Solve for x Jeff Bivin -- LZHS

80. Solve for x Jeff Bivin -- LZHS

81. Solve for x Jeff Bivin -- LZHS

82. Solve for x Jeff Bivin -- LZHS

83. Solve for x Jeff Bivin -- LZHS