Logarithm Functions

1. Section 6-3
L o g a r i t h m s

2. Warm-up
Solve without a calculator.
a. 10 = .0001
a
b. 10 = .01
a
c. 10 = 1
a
d. 10 = 10
a
e. 10 = 100
a
f. 10 = 100,000,000,000
a
g. 10 = 0
a

3. Warm-up
Solve without a calculator.
a. 10 = .0001
a
b. 10 = .01
a
c. 10 = 1
a
a = −4
d. 10 = 10
a
e. 10 = 100
a
f. 10 = 100,000,000,000
a
g. 10 = 0
a

4. Warm-up
Solve without a calculator.
a. 10 = .0001
a
b. 10 = .01
a
c. 10 = 1
a
a = −4 a = −2
d. 10 = 10
a
e. 10 = 100
a
f. 10 = 100,000,000,000
a
g. 10 = 0
a

5. Warm-up
Solve without a calculator.
a. 10 = .0001
a
b. 10 = .01
a
c. 10 = 1
a
a = −4 a = −2 a=0
d. 10 = 10
a
e. 10 = 100
a
f. 10 = 100,000,000,000
a
g. 10 = 0
a

6. Warm-up
Solve without a calculator.
a. 10 = .0001
a
b. 10 = .01
a
c. 10 = 1
a
a = −4 a = −2 a=0
d. 10 = 10
a
e. 10 = 100
a
a= 1
2
f. 10 = 100,000,000,000
a
g. 10 = 0
a

7. Warm-up
Solve without a calculator.
a. 10 = .0001
a
b. 10 = .01
a
c. 10 = 1
a
a = −4 a = −2 a=0
d. 10 = 10
a
e. 10 = 100
a
a= 1
2
a=2
f. 10 = 100,000,000,000
a
g. 10 = 0
a

8. Warm-up
Solve without a calculator.
a. 10 = .0001
a
b. 10 = .01
a
c. 10 = 1
a
a = −4 a = −2 a=0
d. 10 = 10
a
e. 10 = 100
a
a= 1
2
a=2
f. 10 = 100,000,000,000
a
g. 10 = 0
a
a = 11

9. Warm-up
Solve without a calculator.
a. 10 = .0001
a
b. 10 = .01
a
c. 10 = 1
a
a = −4 a = −2 a=0
d. 10 = 10
a
e. 10 = 100
a
a= 1
2
a=2
f. 10 = 100,000,000,000
a
g. 10 = 0
a
a = 11 No solution

10. Definition of
Logarithm

11. Definition of
Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:

12. Definition of
Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
y = log b x IFF b = x y

13. Definition of
Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
y = log b x IFF b = x y
What does this mean?

14. Definition of
Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
y = log b x IFF b = x y
What does this mean?
y = log b x IFF b = x y

15. Definition of
Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
y = log b x IFF b = x y
What does this mean?
y = log b x IFF b = x y
Base

16. Definition of
Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
y = log b x IFF b = x y
What does this mean?
y = log b x IFF b = x y
Base
Exponent

17. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36

18. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
−1

19. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
−1
Why?

20. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
−1
Why?
−1
6 = 1
6

21. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
−1 2
Why?
−1
6 = 1
6

22. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
−1 2
Why? Why?
−1
6 = 1
6

23. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
−1 2
Why? Why?
−1
6 = 1
6 6 = 36
2

24. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
2
−1 2
5
Why? Why?
−1
6 = 1
6 6 = 36
2

25. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
2
−1 2
5
Why? Why? Why?
−1
6 = 1
6 6 = 36
2

26. Example 1
Evaluate.
1
a. log 6 6
b. log 6 36 5
c. log 6 36
2
−1 2
5
Why? Why? Why?
−1
6 = 1 2
6 6 = 36
2
6 = 36 = 6
5 5 5 2

27. Example 2
Evaluate.
log 9 243

28. Example 2
Evaluate.
log 9 243
9 = 81
2

29. Example 2
Evaluate.
log 9 243
9 = 81
2
9 = 729
3

30. Example 2
Evaluate.
log 9 243
9 = 81
2 x is somewhere in between 9 = 729
3

31. Example 2
Evaluate.
log 9 243
9 = 81
2 x is somewhere in between 9 = 729
3
What do we know about 243?

32. Example 2
Evaluate.
log 9 243
9 = 81
2 x is somewhere in between 9 = 729
3
What do we know about 243?
5
243 = 3

33. Example 2
Evaluate.
log 9 243
9 = 81
2 x is somewhere in between 9 = 729
3
What do we know about 243?
1
5
243 = 3 = 9 2

34. Example 2
Evaluate.
log 9 243
9 = 81
2 x is somewhere in between 9 = 729
3
What do we know about 243?
1
5
243 = 3 = 9 2
Ok, what does that mean?

35. Example 2
Evaluate.
log 9 243
9 = 81
2 x is somewhere in between 9 = 729
3
What do we know about 243?
1
5
243 = 3 = 9 2
Ok, what does that mean?
(9 ) = 243
1 5
2

36. Example 2
Evaluate.
log 9 243
9 = 81
2 x is somewhere in between 9 = 729
3
What do we know about 243?
1
5
243 = 3 = 9 2
Ok, what does that mean?
(9 ) = 243
1 5
2
log 9 243 = 5
2

37. Common Logarithms

38. Common Logarithms
Logarithms with a base of 10

39. Common Logarithms
Logarithms with a base of 10
You will see this one on your calculator

40. Example 3
Solve to the nearest hundredth.
10 = 73
y

41. Example 3
Solve to the nearest hundredth.
10 = 73
y
Ok, let’s rewrite this as a logarithm.

42. Example 3
Solve to the nearest hundredth.
10 = 73
y
Ok, let’s rewrite this as a logarithm.
log 73 = y

43. Example 3
Solve to the nearest hundredth.
10 = 73
y
Ok, let’s rewrite this as a logarithm.
log 73 = y

44. Example 3
Solve to the nearest hundredth.
10 = 73
y
Ok, let’s rewrite this as a logarithm.
log 73 = y

45. Example 3
Solve to the nearest hundredth.
10 = 73
y
Ok, let’s rewrite this as a logarithm.
log 73 = y
y ≈ 1.86

46. Example 4
Solve log t = 2.9 to the nearest tenth.

47. Example 4
Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.

48. Example 4
Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.
10 2.9
=t

49. Example 4
Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.
10 2.9
=t
t ≈ 794.3

50. Properties of
Logarithms

51. Properties of
Logarithms
Domain is the set of positive real numbers.

52. Properties of
Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.

53. Properties of
Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.

54. Properties of
Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.
The function is strictly increasing.

55. Properties of
Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.
The function is strictly increasing.
As x increases, y has no bound.

56. Properties of
Logarithms

57. Properties of
Logarithms
As x gets smaller and approaches 0, the
values of the function are negative with larger
absolute values.
That means when x is between 0 and 1, the
exponent will be negative.

58. Properties of
Logarithms
As x gets smaller and approaches 0, the
values of the function are negative with larger
absolute values.
That means when x is between 0 and 1, the
exponent will be negative.
The y-axis is an asymptote.

59. Homework

60. Homework
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