1. Chapter 8 Exponential and Logarithmic Functions

2. Exponential Functions

3. Exponential Functions
An exponential function is a function with the general form
y = abx
a ≠ 0 and b > 0, and b ≠ 1
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 4. y = 7( 2)x
5. y = 2( 1.25 )x 6. 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 a model
Write an exponential model for a graph that includes the points
(2,122.5) and (3,857.5)
(0,24) and (3, 8/9)

8. Writing Exponential Functions
Cooling times for a cup of coffee at various temps.

9. 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?

10. 2(3)

11. 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.

12. Exponential Functions

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

14. 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?

15. 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?

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

17. 50(1/2)^(1/14)x

18. Increase and Decrease by percent
Exponential Models can also be used to show an increase or decrease by a percentage.

19. Increase and Decrease by percent
The rate of increase or decrease is a percent, we use a change factor/base
of 1 + r or 1 – r.
Growth Decay
b > < b < 1
Change factor Change Factor
(1 + r) (1 - r)

20. Percent to Change Factor
When something grows or decays by a percent, we have to add or subtract it from one to find b.
Increase of 25% 2. increase of 130%
Decrease of 30% 4. Decrease of 80%

21. 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

22. 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?

23. 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?

24. Compound Interest
Compound Interest is another type exponential function: Here P = starting amount R = rate n = period T = time

25. Compound Interest
Find the balance of a checking account that has $3,000 compounded annually at 14% for 4 years.

26. Compound Interest
Find the balance of a checking account that has $500 compounded semiannually at 8% for 5 years.

27. 8.3 Logarithmic Functions

28. Logarithmic Expressions
Solve for x:
2x = 4
2x = 16
2x = 10

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

30. Exp to Log form
Given the following Exponential Functions, Convert to Logarithmic Functions.
42 = = 5
3. 70 = 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. Try Some!

34. Common Log
A Common Logarithm is a logarithm that uses base 10. Log 10 y = x > Log y = x EX. Log1000

35. Common Log
The Calculator will do a Common Log for us!! Find the Log Log1000 Log100 Log(1/10)

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

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

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

39. Properties of Logarithms

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

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

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

43. Expand Each Logarithm
Log 5 (x/y)
Log 3r4
Log 2 7b

44. 8.5 and Logarithmic Equations

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

46. Exponential Equations
An Exponential Equation is an equation with an unknown for an exponent. Ex: 4x = 34

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

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

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

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

51. Try Some!
Log 6 – log 3x = -2
Log 2x + Log x = 11
Log 5 – Log 2x = 1

52. Homework Change!!
PG 456 # 34 – 46 even

53. Classwork
PG 112 # 11 – 30 # 41 – 49 Pick 12 (4 from each section)

54. 8.6 Natural Logarithms

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

56. 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

57. Exponential Functions with e
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.

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

61. Natural Logarithmic
We call a log with a base of 10 “Common Log” We can call a log with a base of e “Natural Log” Natural Log is denoted with the “LN” All the same rules and properties apply to natural log as they do to regular logs

62. Exponential to Log form

63. Log to Exponential Form
Ln 1 = 0
Ln 9 = 2.197
Ln (5.28) =

64. Simplify
3 Ln 5
Ln 5 + Ln 4
Ln 20 – Ln 10
4 Ln x + Ln y – 2 Ln z

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

66. Solving Exponential Equations

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

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