1. 4.2 Logarithmic Functions
2 Corinthians 4:16-18 So we do not lose heart. Though our
outer self is wasting away, our inner self is being renewed day
by day. For this light momentary affliction is preparing for us
an eternal weight of glory beyond all comparison, as we look
not to the things that are seen but to the things that are
unseen. For the things that are seen are transient, but the
things that are unseen are eternal.

2. Introduction: We will look at a “made-up”
function called “box”.

3. Introduction: We will look at a “made-up”
function called “box”.
W
Example: 2 (8) = 3

4. Introduction: We will look at a “made-up”
function called “box”.
W
Example: 2 (8) = 3
3
2 − box of 8 is 3 because 2 = 8

5. Introduction: We will look at a “made-up”
function called “box”.
W
Example: 2 (8) = 3
3
2 − box of 8 is 3 because 2 = 8
so ... we are finding an exponent

6. Example
W
2 (16 ) =

7. Example
W
2 (16 ) = 4

8. Example
W
2 (16 ) = 4
Example
W
2 (2) =

9. Example
W
2 (16 ) = 4
Example
W
2 (2) = 1

10. Example
W
2 (16 ) = 4
Example
W
2 (2) = 1
Example
W
3 ( 27 ) =

11. Example
W
2 (16 ) = 4
Example
W
2 (2) = 1
Example
W
3 ( 27 ) = 3

12. Example
W
3 (9) =

13. Example
W
3 (9) = 2

14. Example
W
3 (9) = 2
Example
W
5 (125 ) =

15. Example
W
3 (9) = 2
Example
W
5 (125 ) = 3

16. Example
W
3 (9) = 2
Example
W
5 (125 ) = 3
Example
W⎛ 1 ⎞
2 ⎜ ⎟ =
⎝ 2 ⎠

17. Example
W
3 (9) = 2
Example
W
5 (125 ) = 3
Example
W⎛ 1 ⎞
2 ⎜ ⎟ = −1
⎝ 2 ⎠
−11 1
because 2 = 1 =
2 2

18. Example
2 ( 2) =
W

19. Example
1
2 ( )
W
2 =
2
1
because 2 = 2
2

20. Example
1
2 W
( )2 =
2
1
because 2 = 2
2
Example
W⎛ 1 ⎞
3 ⎜ ⎟ =
⎝ 9 ⎠

21. Example
1
2 W
( )2 =
2
1
because 2 = 2
2
Example
W⎛ 1 ⎞
3 ⎜ ⎟ = −2
⎝ 9 ⎠
1 1
−2
because 3 = 2 =
3 9

22. Example
W
8 (2) =

23. Example
W 1
8 (2) =
3
1
because 8 = 2
3

24. Example
W 1
8 (2) =
3
1
because 8 = 23
Example
W⎛ 1 ⎞
2 ⎜ ⎟ =
⎝ 16 ⎠

25. Example
W 1
8 (2) =
3
1
because 8 = 2
3
Example
W⎛ 1 ⎞
2 ⎜ ⎟ = −4
⎝ 16 ⎠
−4 1 1
because 2 = 4 =
2 16

26. Example
W
2 ( −2 ) =

27. Example
W
2 ( −2 ) = ∅
x
because 2 must be positive

28. The more traditional notation is:
W
log 2 8 = 3 2 (8) = 3
log base two of eight is three

29. The more traditional notation is:
W
log 2 8 = 3 2 (8) = 3
log base two of eight is three
Example
what is log 3 27?

30. The more traditional notation is:
W
log 2 8 = 3 2 (8) = 3
log base two of eight is three
Example
what is log 3 27?
3

31. The more traditional notation is:
W
log 2 8 = 3 2 (8) = 3
log base two of eight is three
Example Example
1
what is log 3 27? what is log 5 ?
25
3

32. The more traditional notation is:
W
log 2 8 = 3 2 (8) = 3
log base two of eight is three
Example Example
1
what is log 3 27? what is log 5 ?
25
3 −2

33. Example
what is log 100?

34. Example
what is log 100?
this is the Common Log and its base is 10

35. Example
what is log 100?
this is the Common Log and its base is 10
2

36. Example
what is log 100?
this is the Common Log and its base is 10
2
and this can be done on your calculator

37. Example
what is log 100?
this is the Common Log and its base is 10
2
and this can be done on your calculator
Example
what is log 9?

38. Example
what is log 100?
this is the Common Log and its base is 10
2
and this can be done on your calculator
Example
what is log 9?
.9542

39. Example
what is ln 82?

40. Example
what is ln 82?
this is the Natural Log and its base is e
and this also can be done on your calculator

41. Example
what is ln 82?
this is the Natural Log and its base is e
and this also can be done on your calculator
4.4067

42. Example
what is ln 82?
this is the Natural Log and its base is e
and this also can be done on your calculator
4.4067
Example
what is ln 16?

43. Example
what is ln 82?
this is the Natural Log and its base is e
and this also can be done on your calculator
4.4067
Example
what is ln 16?
2.7726

44. Def. For all positive real numbers, x and b ,
b > 0, b ≠ 1
y
y = log b x iff x = b

45. Def. For all positive real numbers, x and b ,
b > 0, b ≠ 1
y
y = log b x iff x = b
y
log b x = y ↔ b = x

46. HW #3
I never worked a day in my life. It was all fun!
Thomas Edison