1. Chapter 8 Exponential and Logarithmic Functions

2. 8.1 Exponential Models

3. Exponential Functions
An exponential function is a function with the general form
y = abx
Graphing Exponential Functions
What does a do? What does b do?
1. y = 3( ½ )x  2. y = 3( 2)x
3. y = 5( 2)x y = 7( 2)x
5. y = 2( 1.25 )x y = 2( 0.80 )x

4. A and B
A is the y-intercept B is direction Growth Decay b > 1 0 < b < 1

5. Y-Intercept and Growth vs. Decay
Identify each y-intercept and whether it is a growth or decay.
Y= 3(1/4)x
Y= .5(3)x
Y = (.85)x

6. Writing Exponential Functions
Write an exponential model for a graph that includes the points (2,2) and (3,4). STAT  EDIT STAT  CALC  0:ExpReg

7. Write an exponential model for a graph that includes the points
(2, 122.5) and (3, )
(0, 24) and (3, 8/9)

8. Modeling Exponential Functions
Suppose 20 rabbits are taken to an island. The rabbit population then triples every year.
The function f(x) = 20 • 3x where x is the number of years, models this situation.
What does “a” represent in this problems? “b”?
How many rabbits would there be after 2 years?

9. 2(3)

10. Intervals
When something grows or decays at a particular interval, we must multiply x by the intervals’ reciprocal. EX: Suppose a population of 300 crickets doubles every 6 months. Find the number of crickets after 24 months.

11. 8.2 Exponential Functions

12. Exponential Function
Where a = starting amount (y – intercept) b = change factor x = time

13. Modeling Exponential Functions
Suppose a Zombie virus has infected 20 people at our school. The number of zombies doubles every hour. Write an equation that models this. How many zombies are there after 5 hours?

14. Modeling Exponential Functions
Suppose a Zombie virus has infected 20 people at our school. The number of zombies doubles every 30 minutes. Write an equation that models this. How many zombies are there after 5 hours?

15. A population of 2500 triples in size every 10 years.
What will the population be in 30 years?

16. 50(1/2)^(1/14)x

17. Growth Decay . b > 1 0 < b < 1 (1 + r) (1 - r)

18. Percent to Change Factor
Increase of 25% Increase of 130%
Decrease of 30% Decrease of 80%

19. Growth Factor to Percent
Find the percent increase or decease from the following exponential equations.
y = 3(.5)x
y = 2(2.3)x
y = 0.5(1.25)x

20. Percent Increase and Decrease
A dish has 212 bacteria in it. The population of bacteria will grow by 80% every day.
How many bacteria will be present in 4 days?

21. Percent Increase and Decrease
The house down the street has termites in the porch. The exterminator estimated that there are about 800,000 termites eating at the porch. He said that the treatment he put on the wood would kill 40% of the termites every day.
How many termites will be eating at the porch in 3 days?

22. Compound Interest
P = starting amount R = rate n = period T = time

23. Compound Interest
P =
R =
n =
T =
Find the balance of a checking account that has $3,000 compounded annually at 14% for 4 years.

24. Compound Interest
P =
R =
n =
T =
Find the balance of a checking account that has $500 compounded semiannually at 8% for 5 years.

25. Warm Up
Check your homework answers and write ones that you need to see done on the board. 1) Decay, y-int is 1, decrease of 38% Growth, y-int is 3, increase of 19% 2) 1st bounce = 90 inches, 5th bounce = 28.5, difference of 61.5 inches 3) a) $12, b) 44 years 4) $17, ) .625 mg 6) 53, 144, 100

26. QUIZ!!

27. 8.3 Logarithmic Functions

28. Logarithmic Expressions
Solve for x:
2x = 4
x = 10

29. Logarithmic Expression
A Logarithm solves for the missing exponent:
Exponential Form
Logarithmic Form
y = bx
logby = x

30. Convert the following exponential functions to logarithmic Functions.
= 16
= 5
= 1

31. Log to Exp form
Given the following Logarithmic Functions, Convert to Exponential Functions. 1. Log4 (1/16) = Log255 = ½

32. Evaluating Logarithms
To evaluate a log we are trying to “find the exponent.” Ex: Log5 25 Ask yourself: 5x = 25

33. You Try!

34. A Common Logarithm is a logarithm that uses base 10
A Common Logarithm is a logarithm that uses base 10. log 10 y = x ---- > log y = x Example: log1000

35. Common Log Find the Log: Log100 Log(1/10)
The Calculator will do a Common Log for us!
Find the Log:
Log100
Log(1/10)

36. When the base of the log is not 10, we can use a Change of Base Formula to find Logs with our calculator:

37. You Try!
Find the following Logarithms using change of base formula

38. Graph the pair of equations
y = 2x and y = log 2 x
y = 3x and y = log 3 x
What do you notice??

39. Graphing Logarithmic Functions
A logarithmic function is the inverse of an exponential function. The inverse of a function is the same as reflecting a function across the line y = x

40. 8.4 Properties of Logarithms

41. Properties of Logs Product Property loga(MN)=logaM + logaN
Quotient Property
loga(M/N)=logaM – logaN
Power Property
Loga(Mp)=p*logaM

42. Identify the Property Log 2 8 – log 2 4 = log 2 2
Log b x3y = 3(log b x) + log b y

43. Simplify Each Logarithm
Log 3 20 – log 3 4
3(Log 2 x) + log 2 y
3(log 2) + log 4 – log 16

44. Warm Up - Expand Each Logarithm
Log 5 (x/y)
Log 3r4
Log 2 7b
This is in your notes from yesterday

45. HW Check – Properties of Logarithms Worksheet

46. Classwork – (same as HW worksheet)
Problems 27, 30, 32, 34, 36, 40, 42

47. 8.5 Exponential and Logarithmic Equations

48. Remember!
Exponential and Logarithmic equations are INVERSES of one another. Because of this, we can use them to solve each type of equation!

49. Exponential Equations
To solve for a variable in an exponent, take the log of each side. Ex: 4x = 34

50. Try Some!
5x = 27
4. 3x+4 = 101
2. 73x = 20
5. 11x = 250
3. 62x = 21

51. Logarithmic Equation
To Solve Logarithmic Equation we can transform them into Exponential Equations! Ex: Log (3x + 1) = 5

52. You Try!
Log (7 – 2x) = -1
Log ( 5 – 2x) = 0
Log (6x) – 3 = -4

53. Using Properties to Solve Equations
Use the properties of logs to simplify logarithms first before solving! Ex: 2 log(x) – log (3) = 2

54. You Try!
log 6 – log 3x = -2
log 5 – log 2x = 1

55. 8.6 Natural Logarithms

56. Compound Interest
Find the balance in an account paying 3.2% annual interest on $10,000 in 18 years compounded quarterly.

57. The Constant: e
e is a constant very similar to π. Π = … e = … Because it is a fixed number, we can find: e2 e3 e4

58. Exponential Functions with a base of e are used to describe CONTINUOUS growth or decay. Some accounts compound interest, every second. We refer to this as continuous compounding.

59. Continuously Compounded
Find the balance in an account paying 3.2% annual interest on $10,000 in 18 years compounded continuously.

60. Investment: You put $2000 into an account earning 4% interest compounded continuously. Find the amount at the end of 8 years.

61. If $5,000 is invested in a savings account that pays 7
If $5,000 is invested in a savings account that pays 7.85% interest compounded continuously, how much money will be in the account after 12 years?

62. Natural Logarithms
Log with a base of 10: “Common Log”
Log with a base of e: “Natural Log” (ln)
- The natural logarithm of a number x is the power to which e would have to be raised to equal x
Note: All the same rules and properties apply to natural log as they do to regular logs

63. Exponential to Log form

64. Log to Exponential Form
ln 1 = 0
ln 9 = 2.197
ln (5.28) =

65. Simplify
3 ln 5
ln 5 + ln 4
ln 20 – ln 10
4 ln x + ln y – 2 ln z

66. Expand
Ln (xy2)
Ln(x/4)
Ln(y/2x)

67. Solving Exponential Equations

68. Solving Logarithmic Equations
Ln x = -2
Ln (2m + 3) = 8
1.1 + Ln x2 = 6

69. Homework
PG 464 # 2 – 8, 14 – 28 (all even)