1. 5.1 Exponential Functions
Rules For Exponents
If a > 0 and b > 0, the following hold true for all real
numbers x and y.

2. 5.1 Exponential Functions
If we apply the quotient rule, we get:

3. 5.1 Exponential Functions
For any nonzero number x:
and

4. 5.1 Exponential Functions
Examples:

5. 5.1 Exponential Functions
Examples:
(5x-2)3 = 125x-6=125/x6
(3x/y3)2 = 9x2/y6
(4x)-1 = 1/(4x)
(2a3b-3c4)3 = 8a9b-9c12
40 = 1
2-1 = ½
(½)-2 = 4
5-2 = 1/25

6. 5.1 Exponential Functions
Simplify:

7. 5.1 Exponential Functions

8. 5.1 Exponential Functions
Simplify:
Rewrite:
Notice:

9. 5.1 Exponential Functions
Simplify:
Rewrite:
Notice:

10. 5.1 Exponential Functions
Simplify:
Rewrite:
Notice:

11. 5.1 Exponential Functions
Then If
Examples
Since
Since
Since

12. 5.1 Exponential Functions
In general, if n is a multiple of m, then
If n is odd
If n is even

13. 5.1 Exponential Functions
Use the rules for exponents to
solve for x
4x = 128
(2)2x = 27
2x = 7
x = 7/2
2x = 1/32
2x = 2-5
x = -5

14. 5.1 Exponential Functions
27x = 9-x+1
(33)x = (32)-x+1
33x = 3-2x+2
3x = -2x+ 2
5x = 2
x = 2/5
(x3y2/3)1/2
x3/2y1/3

15. 5.1 Exponential Functions
Definition Exponential Function
Let a be a positive real number other than 1,
the function f(x) = ax is the exponential
function with base a.

16. 5.1 Exponential Functions4 3 2 1 -2 -3 -4 -5 y x
If b > 1, then the graph of b x will:
Rise from left to right.
Not intersect the x-axis.
Approach the x-axis.
Have a y-intercept of (0, 1)
y = 2 x

17. 5.1 Exponential Functions4 3 2 1 -2 -3 -4 -5 y x
If 0 < b < 1, then the graph of b x will:
Fall from left to right.
Not intersect the x-axis.
Approach the x-axis.
Have a y-intercept of (0, 1)
y = (1/2) x

18. 5.1 Exponential Functions
Natural Exponential Function where e is the natural base and e  2.718…

19. 5.1 Exponential Functions
f(x) = 2x
h(x) = (0.5)x
g(x) = ex
Domain
Range
Increasing or Decreasing
Point Shared On All Graphs
(-∞, ∞)
(-∞, ∞)
(-∞, ∞)
(0, ∞)
(0, ∞)
(0, ∞)
Inc.
Dec.
Inc.
(0, 1)

20. 5.1 Exponential Functions
Use translation of functions to graph the following Determine the domain and range
f (x) = 2(x + 2) – 3
Domain (-∞, ∞)
Range (-3, ∞)

21. 5.1 Exponential Functions
Definitions Exponential Growth and Decay
The function y = k ax, k > 0 is a model for
exponential growth if a > 1, and a model
for exponential decay if 0 < a < 1.
y new amount
yO original amount
b base
t time
h half life

22. 5.1 Exponential Functions
An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.
Find the amount remaining after t hours.
Find the amount remaining after 60 hours.
a. y = yobt/h
y = 2 (1/2)(t/15)
b. y = yobt/h
y = 2 (1/2)(60/15)
y = 2(1/2)4
y = .125 g

23. 5.1 Exponential Functions
A bacteria double every three days. There are 50 bacteria initially present
Find the amount after 2 weeks.
When will there be 3000 bacteria?
a. y = yobt/h
y = 50 (2)(14/3)
y = 1269 bacteria

24. 5.1 Exponential Functions
A bacteria double every three days. There are 50 bacteria initially present
When will there be 3000 bacteria?
b. y = yobt/h
3000 = 50 (2)(t/3)
60 = 2t/3

25. 5.2 Simple and Compound Interest
Formulas for Simple Interest
Suppose P dollars are invested at a simple interest
rate r, where r is a decimal, then P is called the
principal and P ·r is the interest received at the
end of one interest period.

26. 5.2 Simple and Compound Interest
Formulas for Compound Interest
After t years, the balance A in an account with principal
P and annual interest rate r is given by the two
formulas below.
1. For n compoundings per year:
2. For continuous compounding:

27. 5.2 Simple and Compound Interest
Find the balance after 10 years if $ is invested
at 4% and the account pays simple interest.

28. 5.2 Simple and Compound Interest
Find the balance after 10 years if $ is invested
at 4% and the interest is compounded:
a. Semiannually $
b. Monthly: $
c. Continuously: $

29. 5.3 Effective Rate and Annuities
Effective Annual Rate
The effective annual rate of ieff of APR
compounded k times per year is given by the
equation
Another name for effective annual rate is
effective yield

30. 5.3 Effective Rate and Annuities
What is the better rate of return,
7% compounded quarterly or 7.2 %
compounded semianually?

31. 5.3 Effective Rate and Annuities
1.071 – 1 =
.071 = 7.1%
1.073 – 1 =
.073 = 7.3%
7.2% compounded semiannually is better.

32. 5.3 Effective Rate and Annuities
What is the better rate of return,
8 % compounded monthly or 8.2 %
compounded quarterly?

33. 5.3 Effective Rate and Annuities
8.3%
8.5%
8.2% quarterly is better.

34. 5.3 Effective Rate and Annuities
Future Value of an Ordinary Annuity
The Future Value S of an ordinary annuity
consisting of n equal payments of R dollars,
each with an interest rate i per period is

35. 5.3 Effective Rate and Annuities
Suppose $25.00 per month is invested
at 8% compounded quarterly. How much will
be in the account after one year?
1st quarter $25.00
2nd quarter $25.00(1+.08/4)+ $25.00 = $50.50
3rd quarter $50.50(1+.08/4)+ $25.00 = $76.51
4th quarter $76.51(1+.08/4) + $25.00 = $103.04

36. 5.3 Effective Rate and Annuities
Present Value of an Ordinary Annuity
The Present Value A of an ordinary
annuity consisting of n equal payments of R dollars, each with an interest rate i per period is

37. 5.4 Logarithmic Functions
The inverse of an exponential function is called a logarithmic function.
Definition: x = a y if and only if
y = log a x

38. 5.4 Logarithmic Functions

39. 5.4 Logarithmic Functions
Sketch a graph of f (x) = 2x and sketch a graph of its inverse. What is the domain and range of the inverse of f.
Domain: (0, ∞)
Range: (-∞, ∞)

40. 5.4 Logarithmic Functions
The function f (x) = log a x is called a logarithmic function.
Domain: (0, ∞)
Range: (-∞, ∞)
Asymptote: x = 0
Increasing for a > 1
Decreasing for 0 < a < 1
Common Point: (1, 0)

41. 5.4 Logarithmic Functions
Find the inverse of g(x) = 3x.
(1,3)
(0,1)
(-1,1/3)
(3,1)
Note: The function and
it’s inverse are symmetrical
about the line y = x.
(1,0)
(1/3,-1)

42. 5.4 Logarithmic Functions
Find the inverse of g(x) = ex.
ln x is called the natural
logarithmic function

43. 5.4 Logarithmic FunctionsSo So So So

44. 5.4 Logarithmic Functions
loga(ax) = x for all x  
alog ax = x for all x > 0
loga(xy) = logax + logay
loga(x/y) = logax – logay
logaxn = n logax
Common Logarithm: log 10 x = log x
Natural Logarithm: log e x = ln x
All the above properties hold.

45. 5.4 Logarithmic Functions
Product Rule

46. 5.4 Logarithmic Functions
Quotient Rule

47. 5.4 Logarithmic Functions
Power Rule

48. 5.4 Logarithmic Functions
Expand

49. 5.4 Logarithmic Functions
Find an equation of best fit for the data
(1,3), (2,12), (3,27), (4,48)

50. 5.5 Graphs of Logarithmic Functions
The function f (x) = log a x is called a logarithmic function.
Domain: (0, ∞)
Range: (-∞, ∞)
Asymptote: x = 0
Increasing for a > 1
Decreasing for 0 < a < 1
Common Point: (1, 0)

51. 5.5 Graphs of Logarithmic Functions
The natural and common logarithms can be found on your calculator. Logarithms of other bases are not. You need the change of base formula.
where b is any other appropriate base.
(usually base 10 or base e)

52. 5.5 Graphs of Logarithmic Functions3 5 1 11 2 29
Sketch the graph of
Domain (2,)
Range (-, )

53. 5.5 Graphs of Logarithmic Functions
Sketch the graph of
Domain (-2,)
Range (-, )

54. 5.5 Graphs of Logarithmic Functions
Sketch the graph of
Domain (-3,)
Range (-, )

55. 5.5 Graphs of Logarithmic Functions
On the Richter scale, the magnitude R of an earthquake can be measured by the intensity model.
R = Magnitude
a = Amplitude
T = Period
B = Damping Factor

56. 5.5 Graphs of Logarithmic Functions
What is the magnitude on the Richter scale of an
earthquake if a = 300, T = 30 and B = 1.2?

57. 5.6 Solving Exponential Equations
Solve: 4 3x = 16 x – 2
The bases can be rewritten as:
(22) 3x = (24) (x – 2)
2 6x = 2 4x – 8
6x = 4x – 8
2x = -8
x = -4

58. 5.6 Solving Exponential Equations
To solve exponential equations, pick a convenient base (often base 10 or base e) and take the log of both sides.
Solve:

59. 5.6 Solving Exponential Equations
Take the log of both sides:
Power rule:

60. 5.6 Solving Exponential Equations
Solve for x:
Divide:

61. 5.6 Solving Exponential Equations
To solve logarithmic equations, write both sides of the equation as a single log with the same base, then equate the arguments of the log expressions.
Solve:

62. 5.6 Solving Exponential Equations
Write the left side as a single logarithm:

63. 5.6 Solving Exponential Equations
Equate the arguments:

64. 5.6 Solving Exponential Equations
Solve for x:

65. 5.6 Solving Exponential Equations

66. 5.6 Solving Exponential Equations
Check for extraneous solutions.
x = -3, since the argument of a log cannot be negative

67. 5.6 Solving Exponential Equations
To solve logarithmic equations with one side of the equation equal to a constant, change the equation to an exponential equation
Solve:

68. 5.6 Solving Exponential Equations
Write the left side as a single logarithm:

69. 5.6 Solving Exponential Equations
Write as an exponential equations:

70. 5.6 Solving Exponential Equations
Solve for x: