1. 5.1 Exponential Functions

2. Our presentation today will consists
of two sections.
Section 1: Exploration of exponential functions and their graphs.
Section 2: Discussion of the equality property and its applications.

3. First, let’s take a look at an exponential functiony 1 2 4 -1
1/2 -2
1/4

4. First let’s change the base b to positive values
What conclusion can
we draw ?

5. Next, observe what happens when b assumes a value such that 0<b<1.
Can you explain why
this happens ?

6. will happen if ‘b’ is negative ?
What do you think
will happen if ‘b’ is negative ?

7. Don’t forget our definition !
Any equation of the form:
Can you explain why ‘b’ is restricted from assuming negative values ?

8. To see what impact ‘a’ has on our graph
we will fix the value of ‘b’ at 3.
What does a larger
value of ‘a’
accomplish ?

9. Let’s see if you are correct !
Shall we speculate as to what happens when ‘a’ assumes negative values ?
Let’s see if you are correct !

11. Our general exponential form is
“b” is the base of the function and changes here will result in:
When b>1, a steep increase in the value of ‘y’ as ‘x’ increases.
When 0<b<1, a steep decrease in the value of ‘y’ as ‘x’ increases.

12. We also discovered that changes in “a” would change the y-intercept on its corresponding graph.
Now let’s turn our attention to a useful property of exponential functions.

13. The Equality Property of Exponential Functions
Section 2
The Equality Property of Exponential Functions

14. We know that in exponential functions the exponent is a variable.
When we wish to solve for that variable we have two approaches we can take.
One approach is to use a logarithm. We will learn about these in a later lesson.
The second is to make use of the Equality
Property for Exponential Functions.

15. The Equality Property for Exponential Functions
This property gives us a technique to solve
equations involving exponential functions.
Let’s look at some examples.
Basically, this states that if the bases are the same, then we can simply set the exponents equal.
This property is quite useful when we
are trying to solve equations
involving exponential functions.
Let’s try a few examples to see how it works.

16. Here is another example for you to try:
(Since the bases are the same we
simply set the exponents equal.)
Here is another example for you to try:
Example 1a:

17. The next problem is what to do when the bases are not the same.
Does anyone have
an idea how
we might approach this?

18. Here for example, we know that
Our strategy here is to rewrite the bases so that they are both the same.
Here for example, we know that

19. Example 2: (Let’s solve it now)
(our bases are now the same
so simply set the exponents equal)
Let’s try another one of these.

20. Example 3 Remember a negative exponent is simply
another way of writing a fraction
The bases are now the same
so set the exponents equal.

21. By now you can see that the equality property is
actually quite useful in solving these problems.
Here are a few more examples for you to try.

22. SIMPLE AND COMPOUND INTEREST
Since this section involves what can happen to your money, it should be of INTEREST to you!

23. I = PRT IMPLE INTEREST FORMULA Annual interest rate Interest paid
Time (in years)
Principal (Amount of money invested or borrowed)

24. If you invested $ in an account that paid simple interest, find how long you’d need to leave it in at 4% interest to make $10.00.
enter in formula as a decimal
I = PRT
10 = (200)(0.04)T
1.25 yrs = T
Typically interest is NOT simple interest but is paid semi-annually (twice a year), quarterly (4 times per year), monthly (12 times per year), or even daily (365 times per year).

25. COMPOUND INTEREST FORMULA
annual interest rate (as a decimal)
Principal (amount at start)
time (in years)
amount at the end
number of times per year that interest in compounded

26. Find the amount that results from $500 invested at 8% compounded quarterly after a period of 2 years. 4
(2)
.08
500 4
Effective rate of interest is the equivalent annual simple rate of interest that would yield the same amount as that made compounding. This is found by finding the interest made when compounded and subbing that in the simple interest formula and solving for rate.
Find the effective rate of interest for the problem above.
The interest made was $ Use the simple interest formula and solve for r to get the effective rate of interest.
I = Prt =(500)r(2)
r = = 8.583%

27. 5.2 Natural Exponential Functions Continuous Compound Interest
A = the accumulated amount after t years
P = Principal
r = Nominal interest rate per year
t = Number of years

28. 5.2 Natural Exponential Functions The Natural Base e
Many applied exponential functions involves the irrational number, … symbolized by the letter e.
The number e is called the natural base.
The function f(x) = ex is called the natural exponential function.

29. 5.2
Find the accumulated amount of money after 25 years if $7500 is invested at 12% per year compounded continuously.

30. 5.2
An investment of $10,000 increased to $28, in 15 years. If interest was compounded continuously, find the interest rate.
A = Pert

31. 5.2 Model of Population Growth
If b is the annual birth rate, d is the annual death rate, t is the time (in years), P0 is the initial population at t = 0, and P is the current population, then P = P0ekt where k = b - d is the annual growth rate, the difference between the annual birth rate and death rate.

32. 5.2
The population of a city in 1970 was 153,800. Assuming the population increases continuously at a rate of 5% per year, predict the population of the city in 2010.

33. 5.2
If f(x) = x2(-2e-2x) + 2xe-2x, find the zeros.

34. 5.2 Assignment
pp (1-25 odd)
Transparencies: 19, 21, 23, 25

35. 5.3 Introduction To Logarithms

36. Logarithms were originally developed to simplify complex
arithmetic calculations.
They were designed to transform
multiplicative processes
into additive ones.

37. Definition of Logarithm
Suppose b>0 and b≠1,
there is a number ‘p’
such that:

38. So let’s get a lot of practice with this !
You must be able to convert an exponential equation into logarithmic form and vice versa.
So let’s get a lot of practice with this !

39. We read this as: ”the log base 2 of 8 is equal to 3”.
Example 1:
Solution:
We read this as: ”the log base 2 of 8 is equal to 3”.

40. Read as: “the log base 4 of 16 is equal to 2”.
Example 1a:
Solution:
Read as: “the log base 4 of 16 is equal to 2”.

41. Example 1b:
Solution:

42. Okay, so now it’s time for you to try some on your own.

43. Solution:

44. Solution:

45. Solution:

46. This is simply the reverse of
It is also very important to be able to start with a logarithmic expression and change this into exponential form.
This is simply the reverse of
what we just did.

47. Example 1:
Solution:

48. Example 2:
Solution:

49. Okay, now you try these next three.

50. Solution:

51. Solution:

52. Solution:

53. Let’s rewrite the problem in exponential form.
Solution:
Let’s rewrite the problem in exponential form.
We’re finished !

54. Rewrite the problem in exponential form.
Solution:
Rewrite the problem in exponential form.

55. Solution: Example 3 Try setting this up like this:
Now rewrite in exponential form.

56. Solution: Example 4 First, we write the problem with a variable.
Now take it out of the logarithmic form
and write it in exponential form.

57. Solution: Example 5 First, we write the problem with a variable.
Now take it out of the exponential form
and write it in logarithmic form.

58. Basically, with logarithmic functions,
Finally, we want to take a look at the Property of Equality for Logarithmic Functions.
Basically, with logarithmic functions,
if the bases match on both sides of the equal sign , then simply set the arguments equal.

59. Since the bases are both ‘3’ we simply set the arguments equal.
Example 1
Solution:
Since the bases are both ‘3’ we simply set the arguments equal.

60. Solution: Example 2 Factor
Since the bases are both ‘8’ we simply set the arguments equal.
Factor
continued on the next page

61. Solution: Example 2 continued
It appears that we have 2 solutions here.
If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution.
Let’s end this lesson by taking a closer look at this.

62. Can anyone give us an explanation ?
Our final concern then is to determine why logarithms like the one below are undefined.
Can anyone give us an explanation ?

63. We’ll then see why a negative value is not permitted.
One easy explanation is to simply rewrite this logarithm in exponential form.
We’ll then see why a negative value is not permitted.
First, we write the problem with a variable.
Now take it out of the logarithmic form
and write it in exponential form.
What power of 2 would gives us -8 ?
Hence expressions of this type are undefined.

64. 5.3 Answers 6. t = 1/4loga10/3 12. a.) 10-8 = x; b.) 10y-2 = x
c.) e1/2 =x d.) e7+x = z e.) e1.2 = t-5
18. -3/ a) 78.8
b) 92,900 c)
36. graph d) 40.4
e) 2.59 f)

65. 5.4 Properties of logarithms
Objectives
Use the product rule.
Use the quotient rule.
Use the power rule.
Expand logarithmic expressions.
Condense logarithmic expressions.

66. Logarithms of Products
The Product Rule
For any positive numbers M and N and any logarithmic base a,
loga MN = loga M + loga N.
(The logarithm of a product is the sum of the logarithms of the factors.)

67. Example
Condense to a single logarithm:
Expand to a sum of logarithms

68. Logarithms of Powers The Power Rule
For any positive number M, any logarithmic base a, and any real number p,
(The logarithm of a power of M is the exponent times the logarithm of M.)

69. Examples
Expand to a product.
Expand to a product.

70. Examples
Condense the product to a single logarithm using exponents.

71. Logarithms of Quotients
The Quotient Rule
For any positive numbers M and N, and any logarithmic base a, .
(The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.)

72. Examples Expand to a difference of logarithms.
Condense to a single logarithm.

73. Expand a logarithmic expression
Use properties of logarithms to change one logarithm into a sum or difference of others
Example

74. Expand in terms of sums and differences of logarithms.

75. Example
Condense to a single logarithm.

76. 5.5 Exponential & Logarithmic Equations
Exponential Equations with Like Bases
Exponential Equations with Different Bases
Logarithmic Equations
Change of Base Formulas

77. 5.5 Exponential and Logarithmic Equations
Objective: We will consider various types of exponential and logarithmic equations and their applications.

78. Change of Base Formula
The 2 bases we are most able to calculate logarithms for are base 10 and base e. These are the only bases that our calculators have buttons for.
For ease of computing a logarithm, we may want to switch from one base to another.
The new base, a, can be any integer>1, but we often let a=10 or a=e.

79. Compute What is the log, base 5, of 29?
Does this answer make sense? What power would you raise 5 to, to get 29? A little more than 2! (5 squared is 25, so we would expect the answer to be slightly more than 2.)

80. Rewrite as a quotient of natural logarithms

81. Example Find log6 8 using common logarithms.
Solution: First, we let a = 10, b = 6, and M = 8. Then we substitute into the change-of-base formula:

82. Example We can also use base e for a conversion.
Find log6 8 using natural logarithms.
Solution: Substituting e for a, 6 for b and 8 for M, we have

83. Exponential Equations with Like Bases
In an Exponential Equation, the variable is in the exponent. There may be one exponential term or more than one, like…
If you can isolate terms so that the equation can be written as two expressions with the same base, as in the equations above, then the solution is simple. or

84. Exponential Equations with Like Bases
Example #1 - One exponential expression.
1. Isolate the exponential expression and rewrite the constant in terms of the same base.
2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

85. Exponential Equations with Like Bases
Example #2 - Two exponential expressions.
1. Isolate the exponential expressions on either side of the =. We then rewrite the 2nd expression in terms of the same base as the first.
2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

86. Exponential Equations with Different Bases
The Exponential Equations below contain exponential expressions whose bases cannot be rewritten as the same rational number.
The solutions are irrational numbers, we will need to use a log function to evaluate them. or

87. Exponential Equations with Different Bases
Example #1 - One exponential expression.
1. Isolate the exponential expression.
2. Take the log (log or ln) of both sides of the equation.
3. Use the log rule that lets you rewrite the exponent as a multiplier.

88. Exponential Equations with Different Bases
Example #1 - One exponential expression.
4. Isolate the variable.

89. Exponential Equations with Different Bases
Example #2 - Two exponential expressions.
1. The exponential expressions are already isolated.
2. Take the log (log or ln) of both sides of the equation.
3. Use the log rule that lets you rewrite the exponent as a multiplier on each side..

90. Exponential Equations with Different Bases
Example #2 - Two exponential expressions.
4. To isolate the variable, we need to combine the ‘x’ terms, then factor out the ‘x’ and divide.

91. Logarithmic Equations
In a Logarithmic Equation, the variable can be inside the log function or inside the base of the log. There may be one log term or more than one. For example …

92. Logarithmic Equations
Example 1 - Variable inside the log function.
1. Isolate the log expression.
2. Rewrite the log equation as an exponential equation and solve for ‘x’.

93. Logarithmic Equations
Example 3 - Variable inside the base of the log.
1. Rewrite the log equation as an exponential equation.
2. Solve the exponential equation.

94. Ch. 5 Review Answers