1. Exponential and Logarithmic Functions
Chapter 10
Exponential and Logarithmic Functions

2. Lesson 10.1: Exponential Functions

3. Learning Targets:
• I can graph exponential functions. • I can determine if an exponential function is growth or decay. • I can write an exponential function given two points. • I can solve equations involving exponents.

4. Graphing an exponential function:
y = a∙bx standard form of an exponential function
a = y-intercept (0, a)
b = base
x = exponent

5. Example 1: Graphing Exponential Functions
Sketch the graph of y = 4x and identify its domain and range.
Domain:________
Range:_________

6. Example 2: Graphing Exponential Functions
Sketch the graph of y = 0.7x and identify its domain and range.
Domain:________
Range:_________

7. What type of function is it?
Growth
Decay

8. Example 3: Indicate whether each exponential function is growth or decay.

9. Example 4: Write an exponential function whose graph passes through the given points.
y = a∙bx
(0, -2) and (3, -54)

10. Example 5: Write an exponential function whose graph passes through the given points.
y = a∙bx
(0, 7) and (1, 1.4)

11. Example 6: Write an exponential function whose graph passes through the given points.
y = a∙bx
(0, 3) and (-1, 6)

12. Example 7: Write an exponential function whose graph passes through the given points.
y = a∙bx
(0, -18) and (-2, -2)

13. Remember the Exponent Rules:

14. Example 8: Simplify expressions.b.

15. Example 8: Simplify expressions.c. d.

16. Example 9: Solve equations.
Step 1: Make the bases the same.
Step 2: Set the exponents equal.
Step 3: Solve.

17. Example 10: Solve equations.

18. Example 11: Solve equations.

19. Home Fun
10-1 Worksheet

20. Lesson 10-2: Logarithmic Functions
Objectives: I can….
Convert from logarithmic to exponential
form and vice versa.
Evaluate logarithmic expressions.
Solve logarithmic equations.

21. Definition of Logarithm:
Let b > 0 and b
Then n is the logarithm of m to the base b,
written
logb m = n
if and only if
bn = m

22. Check it out!
Exponential Form Logarithmic Form
means

23. Example 1: Convert to exponential form.b.

24. Example 1 (continued) c. d.

25. Flower Power Root Rule
Power
Flower
Power
Root
Root
Root

26. Example 2: Convert to logarithmic form.b.

27. Example 2 (continued) c. d.

28. Example 3: Evaluate logarithmic expressions.

29. Example 3 (continued) b.

30. A couple of intricate ones…d.

31. Think-Pair-Share! a. b.

32. Example 4: Solve logarithmic equations.

33. Example 4 (continued) b.

34. Example 4 (still continued!)

35. Example 4 (last one!) d.

36. Home Fun
10-2 Worksheet

37. Click the mouse button or press the Space Bar to display the answers.
Transparency 3

38. Transparency 3a

39. Lesson 10-3: Properties of Logs
Learning Targets:
I can use the product and quotient properties of logs.
I can use the power property of logs.
I can solve equations using properties of logs.

40. Properties Product Property: Quotient Property: Power Property:
Example:
Quotient Property:
Example:
Power Property:
Example:

41. Example 1: Solving Equations

42. Example 2: Solving Equations

43. Your Turn : Solve each equation.b.

44. Home
10-3 Worksheet

45. Click the mouse button or press the Space Bar to display the answers.
Transparency 3

46. Transparency 3a

47. Algebra 2A – Lesson 10-4
Common Logarithms

48. Lesson 10-4: Common Logs Learning Targets:
I can find common logarithms.
I can solve logarithmic and exponential equations.
I can use the Change of Base Formula.

49. What is a Common Logarithm?
This logarithm is used so frequently, that it is programmed into our calculators.
We write it as:
log m
Note that we don’t write the base for a common log.

50. Example 1: Find Common Logs with a Calculator
Use a calculator to evaluate each logarithm to four decimal places.
No base labeled, so it must be log10 (the common log).
a) log b) log 0.35

51. This is a useful formula, because now we can rewrite ANY log as log10
Change of Base Formula
This is a useful formula, because now we can rewrite ANY log as log10

52. Example 2: Use the Change of Base Formula
Express each log in terms of common logs.
Then, approximate its value to four decimal places.
a) log b) log2 50

53. Example 3: Use logs to solve equations where the power is the variable
Example 3: Use logs to solve equations where the power is the variable. If necessary, round to four decimal places.
a) Solve: 5x = 62

54. If necessary, round to four decimal places.
You Try: x = 17

55. If necessary, round to four decimal places.
b) Solve: x+1 = 11

56. If necessary, round to four decimal places.
You Try: x-3 = 8

57. Home Fun
10-4 Worksheet

58. Algebra 2B - Chapter 10 Section 5
Natural Logarithms

59. Transparency 5a

60. Lesson 10-6: Exponential Growth and Decay Story Problems
Learning Targets:
I can solve problems involving exponential growth (with doubling)
I can solve problems involving exponential decay (with half-life)

61. Growth and Decay Problems
Doubling Half-life
Growth
Decay
Doubling
Half-life

62. Example 1: Doubling
An experiment begins with 300 bacteria and the population doubles every hour. How many bacteria will there be after:
2 hours?
b) hours?

63. Example 2: Decay Problem
Suppose a car you bought new for $35,000 in 2008 depreciates at a rate of 18% per year.
a. Write an equation for the car’s value x years after 2008.
b. What will the car’s value be after 5 years?

64. Example 3: Growth
A computer engineer is hired for a salary of $70,400. If she gets a 5% raise each year, after how many years will she be making $100,000 or more?

65. Example 4: Half-life Radium-226 has a half-life of 1,620 years.
a) Write an equation for the percent of Radium-226 remaining if there is currently 550 grams after x half-life periods.if there currently 550 grams after x half-life periods.
If you begin with 4 grams of Radium-226, how much will remain after three half-life periods?
c) How many years are equal to three half-life periods of Radium-226?

66. Practice Story Problem 1
The population of a certain strain of bacteria grows according to the formula y = a(2)x, where x is the time in hours.
If there are now 50 bacteria, how many will there be in days?

67. Practice Story Problem 2
The population N of a certain bacteria grows according to the equation N = 200(2)1.4t, where t is the time in hours.
How many bacteria were there at the beginning of the experiment?
b) In how many hours will the number of bacteria reach 100,000?

68. Practice Story Problem 3
In 2001, the population of Lagos, Nigeria was about 7,998,000. Use the population growth rate of 4.06% per year
a. Estimate the population in 2009.
b. In about how many years will the population be over 50,000,000?

69. Practice Story Problem 4
You bought a car for $28,500 in It depreciates at 13% each year?
What is the value of the car in 2018?
b. In how many years will the car depreciate to $5000?

70. Practice Story Problem 5
An isotope of Cesium-137 has a half-life of 30 years.
a. If you start with 20 mg of the substance, how many mg will be left after 90 years?
b. After 120 years?

71. Practice Story Problem 6
In 2010, the population of Australia was 17,800,000. In 2014, the population is now 22,000,000. At what rate is the population growing?

72. Home practice
10-6 Worksheet

73. Lesson 10.5: Natural Logarithms
Algebra 2B
Lesson 10.5: Natural Logarithms

74. Learning Targets: I can understand and use base e.
I can solve base e equations and write equivalent expressions.

75. approaches the value e ≈ 2.71828.
Base e
Euler’s number: e
As n increases,
approaches the value e ≈
“e” is used extensively in finance and business

76. Base e and Natural Log
The functions y = ex and y = ln x are inverse functions.
A couple interesting properties:
Find the e key and the LN key on your calculators.

77. Example 1: Write Equivalent Equations Write an equivalent logarithmic or exponential equation.
a) ex = 23
b) ln x ≈

78. Example 1: Write Equivalent Equations Write an equivalent logarithmic or exponential equation.
c) ex = 6
d) ln x = 2.25

79. Example 2: Evaluate Natural Logarithms
a) eln 21 b)

80. Example 3: Solve Equations

81. Example 3: Solve Equationsb)

82. Pert Formula

83. Example 4: Solve Pert Problems
Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously.
a) What is the balance after 8 years?
b) How long will it take for the balance in your
account to reach at least $2000?

84. Your Turn: Pert Problems
Suppose you deposit $1100 into an account paying 5.5% annual interest, compounded continuously.
 a) What is the balance after 8 years?
b) How long will it take for the balance in your account to reach at least $2000?

85. Home Practice
10-5 Worksheet

86. Click the mouse button or press the Space Bar to display the answers.
Transparency 2

87. Transparency 2a

88. Closure
Solve.
24 g of a substance has a half-life of 18 years. How much of the substance will remain after 72 years?