1. Chapter 12 Exponential and Logarithmic Functions
What is a logarithmic function?

2. Inverse Relations and Functions 12-1
What is the inverse of a function?

3. ACTIVATION: Given 3 What is its additive inverse:
What is its multiplicative inverse:
What do you think of when you hear the term inverse?

4. Inverse of a Function Inverse relation—interchanging the x and y
Inverse function—the inverse of a relation has one y for each and every x
Given G={(1, 3), (2, 4), (6, 3), (7, 7)}
Find the inverse = {(3, 1), (4, 2), (3, 6), (7,7)}

5. Finding the inverse of an equation:
Find the inverse of
y = 4x – 5
What is there to change?
x = 4y – 5
x + 5 = 4y
x + 5 = y-1 4

6. Examples:
y = 5x2 – 4

7. Examples: y-1 = x2 -1 For x > 0
Why is the domain of the inverse restricted?
The domain of the original must be x+1 >0 or x>-1
Since the x and y change for the inverse y>-1
How can this happen?
y-1 = x2 -1
For x > 0

8. Test for symmetry to y = x
If you interchange the x and y and get the original equation the function is symmetric to the line y = x
3x + 3y = 8 4x – 4y = 3
Switch
3y + 3x = 8 4y – 4x = 3
addition is commutative subtraction is not
has symmetry to y = x does not have symmetry to y = x

9. Proving 2 functions are inverses
Means they will be reflected over y = x or
If we find f(x) and then test that in f-1(x) we should arrive back at the original number
f(f-1(x)) =x AND f-1(f(x)) = x
Given f(x) = x3 – 5 and g(x) =
are the functions inverses
Test: f(g(x)) and g(f(x))

10. Page: – 520
4 to 36 by 4’s plus 38, 48 and 52
Homework:

11. Exponential and Logarithmic Functions 12-2
What are exponential and logarithmic functions?

12. ACTIVATION: Graph y = 2x x = 2y What is true of these two functions? x
the graphs show they are inverse but
graphing the second is awkward x y -2 -1 1 2 3 y
.25 .5 1 2 4 8 x
y-1 -2 -1 1 2 3 x
.25 .5 1 2 4 8

13. The log is the inverse of the exponential function
Use your calculator to graph
y = 10x and y = log x

14. The transformation equation:
y = a func( x – h) + k
Will it work with the exponential and log equations
Determines
Up/Down movement
+/ -
Determines
Width and if it is inverted
Determines
Left/Right movement
+/ -

15. Transforming exponential and log functions work with a partner to sketch the graphs

16. Transforming exponential and log functions work with a partner to sketch the graphs
what about
y = (1/2)x
y = (2-1)x
y = 2-x

17. Page:
4 – 24 by 4’s and 26
Homework:

18. Exponential and Logarithmic Functions 12-3
Day 3
Exponential and Logarithmic Functions
How are exponential and logarithmic functions related?
Properties of Logarithmic Functions
What are the basic properties of logarithmic functions?
Exponential and Logarithmic Equations
How do you solve exponential and logarithmic equations?

19. ACTIVATION: Simplify: x2x4 x-3
Since exponential and logarithmic equations are inverses there are some properties which are similar Such as multiplying and addition are related: as are division and subtraction: and raising a power to a power means to multiply

20. Primary Rule of Logarithms
Primary rule of logs: logb x = y becomes x = by allows us to convert from the exponential equation to a logarithmic equation or the other way Try: log 3 12 = x and x = = 3x log4 x = 3

21. What would be true of the following and WHY????
loga a = loga x = 0
NOTE: the log of anything to its own base is 1
when the solution is 0 you are taking the log of 1

22. Examples:
Solve
log5 125 = x logx 81 = 4 log6 216 = x

23. Examples:
Solve
log3 x = -3 log = x log = x

24. Page: worksheet
Homework:

25. Exponential and Logarithmic Functions 12-3
Day 4
Exponential and Logarithmic Functions
How are exponential and logarithmic functions related?
Properties of Logarithmic Functions
What are the basic properties of logarithmic functions?
Exponential and Logarithmic Equations
How do you solve exponential and logarithmic equations?

26. How are the laws for exponents and logarithms related
Let m = loga x
x = am Written in exponent form
xn = ( am )n Raise both sides to the power of n
log a xn = mn Convert back to a logarithmic equation
log a xn = n loga x Substitute for m
When a “power” is raised to a power multiply them.

27. By the same type of proofs
let b = logax and c = logay convert x=ab y = ac multiply xy =abac xy = ab+c take the log of both loga xy =loga ab+c convert loga xy = b + c substitute loga xy = logax + loga y

28. By the same type of proofs

29. Examples:
log log 5 – log 40

30. Page: worksheet
Homework:

31. Exponential and Logarithmic Functions 12-3
Day 4
Exponential and Logarithmic Functions
How are exponential and logarithmic functions related?
Properties of Logarithmic Functions
What are the basic properties of logarithmic functions?
Exponential and Logarithmic Equations
How do you solve exponential and logarithmic equations?

32. Examples
log (x2 -1) – log (x+2) = 1
log (2x -15)
log x =2

33. Example:
log (3x + 2) + log (4x – 1) = 2 log 11

34. Page: worksheet
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35. Lesson 12-8
What are the natural log and the number e? What is the change of base ?

36. ACTIVATION: What is THE primary rule of equations?
—whatever you do to one side you must do to the other.
Given x = solve for x

37. Change of base theorem: used when the base and the number are not powers of the same value.
logb x = y
x = by
log x = log by
log x = y log b
log x = y
log b

38. Examples:
Solve
log3 12 = x log5 25 = x

39. Applications: Exponential growth
Exponential growth follows the model
P = P0 ert
P = final amount
P0 = initial amount
r = rate as a decimal
t = time

40. Applications: Exponential growth
Exponential growth follows the model
P = P0 ert
The population of the US was about 203 million in In 1989, it was about 247 million. Find the growth rate of the population.
247 = 203 er( )

41. Applications: Exponential growth
A particular radioactive isotope has a half-life of 2 years. A scientist has 224 grams on hand. How much of the substance will remain as a radio active isotope after 24 years. ½ P0 = P0 e-r • 2 (this is a two step process) ln ½ = ln e-2r = -2r .346 = r then P = 224 e-(.346) (24) P = .054
or A=P(1 + )nt
A = 224 ( )1•12
A = .054

42. Page: worksheet
Homework: