1. Integers as Exponents
Simplify:

2. Rational Numbers as Exponents
Definition of :
Definition of :
What about: ?

3. Rational Numbers as Exponents
If p and q are integers, with q>0, and b is a positive real number, then:

4. Rational Numbers as Exponents
Simplify:

5. Rational Numbers as Exponents
All properties of exponents we’ve already learned apply to rational exponents as well.
Ex) Re-write as a product of powers:

6. Rational Numbers as Exponents
All properties of exponents we’ve already learned apply to rational exponents as well.
Ex) Simplify:

7. Rational Numbers as Exponents
Ex) Write in exponential form and simplest radical form:

8. Learning Log Summary
LT 1 – I can simplify expressions involving rational exponents.
A rational exponent means…
To multiply bases with rational exponents…

9. Solving Equations with Rational Exponents
Ex) Solve:

10. Solving Equations with Rational Exponents
Ex) Solve:

11. Learning Log Summary
LT 2 – I can solve equations involving rational exponents.
The goal of solving an equation is…
To “undo” a rational exponent on a variable…

12. Closure
Homework:
Pg. 458 ~ 1-35 (Odd), 43, 45

13. Paper-Folding
Using a piece of scrap paper, fold the paper to determine how many rectangles exist (only single rectangles) after each fold. Record data in a table.

14. The Exponential Function
We learned about an expression like
yesterday.
It is also possible that the exponent is an irrational number, like:

15. The Exponential Function
Consider where x can be any rational or irrational number. x y
Where is the value of ?

16. Extending the Laws of Exponents to Irrationals
Ex) Simplify:

17. The Exponential Function
If b>0, the function defined by
is called the exponential function with base b.
We will explore the graph of this function on Thursday!

18. Learning Log Summary
LT 3 – I can define an exponential function and describe the meaning of the variables in the equation
An exponential function is one where…
The base of an exponential function is…

19. Solving Exponential Equations
An exponential equation is an equation in which a variable appears in the exponent position.
A simple example:

20. Solving Exponential Equations
Solve:
Express both sides using the same base.
Set the exponents equal and solve for ‘x’.
Check your solution.

21. Solving Exponential Equations
Solve:
Express both sides using the same base.
Set the exponents equal and solve for ‘x’.
Check your solution.

22. Learning Log Summary
LT 4 – I can solve exponential equations by re-writing necessary bases.
An exponential equation is…
To solve an exponential equation…

23. Closure
Homework:
Pg. 461 ~ 1-29 (Odd)

24. Exponential Function Transformations Activity
Using desmos.com as a graphing tool, graph the examples listed in problems 1 and 2 and answer the questions based on the insights gained from the graphs.

25. Learning Log Summary Revisited
LT 3 – I can define an exponential function and describe the meaning of the variables in the equation
The exponential function can be transformed by…
The effect that b has on the exponential function is…

26. Function Composition
Function composition is a way of performing two functions in one.
Given two functions f(x) and g(x), f(g(x)) means “First apply g(x), then apply f(x).”
This is called the composite of the two functions. f(g(x)) is read aloud as “f of g”.

27. Function Composition

28. Function Composition

29. Function Composition

30. Learning Log Summary
LT 5 – I can compose two functions and simplify the result.
To compose two functions…
f(g(x)) means…

31. Inverse Functions

32. Inverse Functions
How can we verify that f and g “undo” each other for every possible x value?

33. Inverse Functions The functions f and g are inverse functions if:
for all x in the domain of g and for all x in the domain of f.
The inverse of function f is written f -1.

34. Learning Log Summary
LT 6 – I can find the inverse of a function and determine if two functions are inverses using function composition.
To find the inverse of a function…
If two functions are inverses…

35. Inverse Functions

36. Inverse Functions

37. Learning Log Summary
LT 7 – I can describe the graphs of inverse functions and use the Horizontal Line Test appropriately.
The graphs of inverse functions…
If a functions passes the Horizontal Line Test…

38. Closure
Homework:
Pg. 466 ~ 1-14 (All)

39. The Logarithmic Function
Does the function have an inverse function? Why?
How can we find it?

40. The Logarithmic Function
For any point (x,y) on the graph of the exponential function…

41. Learning Log Summary
LT 8 – I can describe the logarithmic function and explain its relationship to the exponential function.
The logarithmic function is…
The connection between an exponential and logarithmic function is…

42. The Logarithmic Function
“log base 2 of a”
If , then
means…
means…
means…
means…
means…

43. The Logarithmic Function
Definition of a Logarithm:
If b and N are positive numbers, :
if and only if

44. The Logarithmic Function
Write each equation in exponential form:

45. The Logarithmic Function
Write each equation in logarithmic form:

46. The Logarithmic Function
Simplify each logarithm:

47. The Logarithmic Function
Solve each equation:

48. Learning Log Summary
LT 9 – I can write exponential equations in logarithmic form (and vice-versa) and solve logarithmic equations.
To solve a logarithmic equation…
To re-write a logarithm expression as an exponential expression…

49. Closure
Homework:
Pg. 470 ~ 1-25 (Odd)

50. LT 1-7 Review
Simplify:

51. LT 1-7 Review
Simplify:

52. LT 1-7 Review
Simplify:

53. LT 1-7 Review
Simplify:

54. LT 1-7 Review
Solve:

55. LT 1-7 Review
Solve:

56. LT 1-7 Review
If and
find

57. LT 1-7 Review
If and
find

58. LT 1-7 Review
If and
find

59. LT 1-7 Review
Determine if the functions are inverses:

60. LT 1-7 Review
Find if

61. Properties of Logarithms
Let b be the base of a logarithmic function and let M and N be positive numbers.

62. Properties of Logarithms
Determine if the statement if true or false.

63. Properties of Logarithms
Ex) Express in terms of
and

64. Properties of Logarithms
Ex) Express in terms of
and

65. Properties of Logarithms
Ex) Express as a single logarithm:

66. Properties of Logarithms
Ex) If and , find:

67. Properties of Logarithms
Ex) If and , find:

68. Properties of Logarithms
Ex) Solve:
Check your solutions!

69. Learning Log Summary
LT 10 – I can utilize the properties of logarithms to re-write and evaluate logarithmic expressions.
The properties of logarithms are…
To re-write a logarithmic expression…

70. Closure
Homework:
Pg. 476 ~ 1-25 (Odd)

71. Solving Exponential Equations
Because of its simplicity, the expression
is called the common logarithm.
It is often written as just

72. Solving Exponential Equations
Ex) Simplify:
Between what two integers must x fall?

73. Solving Exponential Equations
Ex) Determine between which two integers x must fall. Then solve for x:

74. Solving Exponential Equations
Using the Calculator to find Roots:

75. Solving Exponential Equations
Solve:

76. Solving Exponential Equations
Solve:
“Calculation Ready” / Exact vs. Approximate Value

77. Solving Exponential Equations
Determine between what two integers t must fall, then find the approximate value of t.

78. Solving Equations
Note the difference between the equations:

79. The Change of Base Formula
Simplify:

80. The Change of Base Formula

81. Learning Log Summary
LT 11 – I can solve exponential equations by using logarithms and use the Change-of-Base Formula to evaluate a logarithm of any base.
To solve an exponential equation…
The Change of Base Formula is…

82. Closure
Homework:
Pg. 481 ~ 1-37 (Odd)

83. Exponential Growth and Decay
Suppose you invest “P” dollars in an account that earns 8% interest, compounded annually…
Value = (Amt at Beginning of Year) + Interest Earned
= 1.00(Amt at Beginning of Year) + .08(Amt at Beginning of Year)
= 1.08(Amt at Beginning of Year)
Time (years) 1 2 3 … t
Value P

84. Exponential Growth and Decay
How long will it take an investment of $1000 to triple in value if it is invested at an annual rate of 12%, compounded quarterly?

85. Exponential Growth and Decay
A population grows by 50% every year.
Time (years) 1 2 3 … t
Value P
A population decreases by 25% every year.
Time (years) 1 2 3 … t
Value P
A population doubles every three years.
Time (years) 1 2 3 … t
Value P

86. Exponential Growth and Decay
A certain bacteria population doubles in size every 12 hours. By how much (by what factor) will it grow in 2 days?

87. Half-Life
The half-life of a radioactive substance is the amount of time that it takes for exactly half of the original substance to remain.
The half-life of an element is 5 years.
Time (years) 5 10 15 … t
Value P

88. Exponential Growth and Decay
The half-life of Carbon-14 (C-14) is 5730 years. How much of a 10 mg sample will remain after 4500 years?

89. Exponential Growth and Decay
Practice Problems:
$100 is invested at 7.2% interest compounded quarterly. Determine how much the investment is worth after 5 years.
The value of a new $3500 sailboat decreases 10% per year. Find its value after 10 years.
A culture of yeast doubles in size every 20 minutes. The size of the culture now is 80. Find its size in 3 hours.
The radioactive gas radon has a half-life of approximately 3.5 days. About how much of a 100 mg sample will remain after 1 week?

90. Learning Log Summary
LT 12 – I can use exponential and logarithmic functions to model and solve growth and decay problems.
When solving for time in a growth/decay problem…
To write a growth or decay model…

91. Closure
Homework:
Pg. 486 ~ 1-11 (Odd)