20. Packet #13 Exponential and Logarithmic Functions
Math 160
Packet #13
Exponential and Logarithmic Functions

21. Exponential functions are useful modeling certain phenomena, like population growth.

22. 𝒂>𝟏
ex: 𝑓 𝑥 = 2 𝑥

23. 𝟎<𝒂<𝟏
ex: 𝑓 𝑥 = 𝑥

24. Note: 𝑎 is called the base of the exponential function. 𝑎>0 and 𝑎≠1

25. What is the horizontal asymptote. _________ What is the domain
What is the horizontal asymptote? _________ What is the domain? _________ What is the range? _________

26. 𝒚=𝟎
What is the horizontal asymptote? _________ What is the domain? _________ What is the range? _________

27. 𝒚=𝟎
What is the horizontal asymptote? _________ What is the domain? _________ What is the range? _________
(−∞,∞)

28. 𝒚=𝟎
What is the horizontal asymptote? _________ What is the domain? _________ What is the range? _________
(−∞,∞)
(𝟎,∞)

29. Natural Base 𝒆
𝑒 (called the natural base) is defined to be the number that 𝑛 𝑛 approaches as 𝑛 gets larger and larger.

30. Natural Base 𝒆
𝑒 (called the natural base) is defined to be the number that 𝑛 𝑛 approaches as 𝑛 gets larger and larger. 𝑛
1+ 1 𝑛 𝑛 1 2
2.25 5 10
100
1,000
1,000,000

31. From the compound interest formula 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 , we can interpret this as investing $1 at a rate of 100% per year for 1 year, compounded 𝑛 times per year.

32. Continuing, we get that 𝑒 is the irrational number 2.718281828459045…

33. Notes: The graph of 𝑒 𝑥 is between the graphs of 2 𝑥 and 3 𝑥
Notes: The graph of 𝑒 𝑥 is between the graphs of 2 𝑥 and 3 𝑥 . (Since 2<𝑒<3.) In calculus, you learn that 𝑒 𝑥 has some amazing properties…

34. Notes: The graph of 𝑒 𝑥 is between the graphs of 2 𝑥 and 3 𝑥
Notes: The graph of 𝑒 𝑥 is between the graphs of 2 𝑥 and 3 𝑥 . (Since 2<𝑒<3.) In calculus, you learn that 𝑒 𝑥 has some amazing properties…

35. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

36. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

37. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

38. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

39. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

40. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

41. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

42. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

43. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

44. Ex 1. Graph 𝑓 𝑥 = 2 𝑥−2 +1

45. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

46. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

47. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

48. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

49. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

50. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

51. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

52. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

53. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

54. Ex 2. Graph 𝑔 𝑥 =− 𝑒 −𝑥

55. One-to-one functions have inverses
One-to-one functions have inverses. The inverse of an exponential function is a logarithmic function. In general, the logarithmic function with base 𝒂 is 𝒇 𝒙 = 𝐥𝐨𝐠 𝒂 𝒙 (where 𝑎>0, 𝑎≠1, 𝑥>0).

56. 𝒂>𝟏
ex: 𝑓 𝑥 = log 3 𝑥

57. 𝟎<𝒂<𝟏
ex: 𝑓 𝑥 = log 𝑥

58. What is the vertical asymptote. _________ What is the domain
What is the vertical asymptote? _________ What is the domain? _________ What is the range? _________

59. 𝒙=𝟎
What is the vertical asymptote? _________ What is the domain? _________ What is the range? _________

60. 𝒙=𝟎
What is the vertical asymptote? _________ What is the domain? _________ What is the range? _________
(𝟎,∞)

61. 𝒙=𝟎
What is the vertical asymptote? _________ What is the domain? _________ What is the range? _________
(𝟎,∞)
(−∞,∞)

62. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

63. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

64. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

65. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

66. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

67. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

68. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

69. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

70. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

71. Ex 3. Graph 𝑓 𝑥 =3− log 2 𝑥

72. Note: 𝐥𝐨𝐠 𝒙= 𝐥𝐨𝐠 𝟏𝟎 𝒙 (_____________ logarithm) 𝐥𝐧 𝒙= 𝐥𝐨𝐠 𝒆 𝒙 (_____________ logarithm)

73. Note: 𝐥𝐨𝐠 𝒙= 𝐥𝐨𝐠 𝟏𝟎 𝒙 (_____________ logarithm) 𝐥𝐧 𝒙= 𝐥𝐨𝐠 𝒆 𝒙 (_____________ logarithm)
common

74. Note: 𝐥𝐨𝐠 𝒙= 𝐥𝐨𝐠 𝟏𝟎 𝒙 (_____________ logarithm) 𝐥𝐧 𝒙= 𝐥𝐨𝐠 𝒆 𝒙 (_____________ logarithm)
common
natural

75. Ex 4. Evaluate. log 2 8= log 3 3= log 𝑒 1 𝑒 3 = log 36 6= ln 𝑒 3 = log 1000 =

76. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____)

77. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____) 𝟏

78. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____) 𝟏 𝟎

79. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____) 𝟏 𝟎

80. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____) 𝟏 𝟎 𝒙

81. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____) 𝟏 𝟎 𝒙 𝒙

82. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____) 𝟏 𝟎 𝒙 𝒙

83. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____) 𝟏 𝟎 𝒙 𝒙 𝟖

84. Note: 𝐥𝐨𝐠 𝒂 𝒂= ____ and 𝐥𝐨𝐠 𝒂 𝟏= ____ Note: Since 𝑎 𝑥 and log 𝑎 𝑥 are inverse functions by definition, 𝒂 𝐥𝐨𝐠 𝒂 𝒙 = _____ and 𝐥𝐨𝐠 𝒂 𝒂 𝒙 = _____ (ex: log = _____ 3 log 3 7 = _____) 𝟏 𝟎 𝒙 𝒙 𝟖 𝟕