Download presentation

Presentation is loading. Please wait.

1
Basic Logarithms A way to Undo exponents

2
Many things we do in mathematics involve undoing an operation.

3
Subtraction is the inverse of addition

4
When you were in grade school, you probably learned about subtraction this way. 2 + = 8 7 + = 10

5
Then one day your teacher introduced you to a new symbol ─ to undo addition

6
3 + = 10 Could be written 10 ─ 3 =

7
8 – 2 =

8
2 + ? = 8

9
8 – 2 = 2 + 6 = 8

10
8 – 2 = 6 2 + 6 = 8

11
The same could be said about division ÷

12
40 ÷ 5 =

13
5 x ? = 40

14
40 ÷ 5 = 5 x 8 = 40

15
40 ÷ 5 = 8 5 x 8 = 40

16
Consider √49

17
√49 = ?

18
? 2 = 49

19
√49 = ? 7 2 = 49

20
√49 = 7 7 2 = 49

21
Exponential Equations: 5 ? = 25

22
Exponential Equations: 5 2 = 25

23
Logarithmic Form of 5 2 = 25 is log 5 25 = 2

24
log 5 25 = ?

25
5 ? = 25

26
log 5 25 = ? 5 2 = 25

27
log 5 25 = 2 5 2 = 25

28
Try this one…

29
log 7 49 = ?

30
7 ? = 49

31
log 7 49 = ? 7 2 = 49

32
log 7 49 = 2 7 2 = 49

33
and this one…

34
log 3 27 = ?

35
3 ? = 27

36
log 3 27 = ? 3 3 = 27

37
log 3 27 = 3 3 3 = 27

38
Remember your exponent rules? 7 0 = ? 5 0 = ?

39
Remember your exponent rules? 7 0 = 1 5 0 = ?

40
Remember your exponent rules? 7 0 = 1 5 0 = 1

41
log 7 1 = ?

42
7 ? = 1

43
log 7 1 = ? 7 0 = 1

44
log 7 1 = 0 7 0 = 1

45
Keep going…

46
log 3 1 = ?

47
3 ? = 1

48
log 3 1 = ? 3 0 = 1

49
log 3 1 = 0 3 0 = 1

50
Remember this? 1/25 = 1/ 5 2 = 5 -2

51
log 5 ( 1/25 )= ?

52
5 ? = 1/25

53
log 5 ( 1/25 )= ? 5 -2 = 1/25

54
log 5 ( 1/25 )=-2 5 -2 = 1/25

55
Try this one…

56
log 3 ( 1/81 )= ?

57
3 ? = 1/81

58
log 3 ( 1/81 )= ? 3 -4 = 1/81

59
log 3 ( 1/81 )=-4 3 -4 = 1/81

60
Let’s learn some new words. When we write log 5 125 5 is called the base 125 is called the argument

61
When we write log 2 8 The base is ___ The argument is ___

62
When we write log 2 8 The base is 2 The argument is ___

63
When we write log 2 8 The base is 2 The argument is 8

64
Back to practice…

65
log 10 1000=?

66
10 ? =1000

67
log 10 1000=? 10 3 =1000

68
log 10 1000=3 10 3 =1000

69
And another one

70
log 10 ( 1/100 )=?

71
10 ? =1/100

72
log 10 ( 1/100)=? 10 -2 =1/100

73
log 10 ( 1/100)=-2 10 -2 =1/100

74
log 10 is used so much that we leave off the subscript (aka base)

75
log 10 100 can be written log 100

76
log 10000 =?

77
10 ? =10000

78
log 10000 =? 10 4 =10000

79
log 10000 = 4 10 4 =10000

80
And again

81
log 10 = ?

82
10 ? =10

83
log 10 = ? 10 1 =10

84
log 10 = 1 10 1 =10

85
What about log 33?

86
What about log 33? We know 10 1 = 10 and 10 2 = 100 since 10 < 33 < 100 we know log 10 < log 33 < log 100

87
Add to log 10 < log 33 < log 100 the fact that log 10 = 1 and log 100 = 2 to get 1 < log 33 < 2

88
A calculator can give you an approximation of log 33. Look for the log key to find out… (okay, get it out and try)

89
log 33 is approximately 1.51851394

90
Guess what log 530 is close to.

91
100 < 530 < 1000 so log 100 < log 530 < log 1000 and thus 2 < log 530 < 3

92
Your calculator will tell you that log 530 ≈ 2.72427….

93
Now for some practice with variables. We’ll be solving for x.

94
log 4 16 = x

95
log 4 16 = x 4 ? = 16

96
log 4 16 = x 4 2 = 16

97
log 4 16 = x x=2 4 2 = 16

98
Find x in this example.

99
log 8 x = 2

100
log 8 x = 2 8 2 = ?

101
log 8 x = 2 8 2 = 64

102
log 8 x = 2 x=64 8 2 = 64

103
We need some rules since we want to stay in real number world. Consider log base (argument) = number The base must be > 0 The base cannot be 1 The argument must be > 0

104
Why can’t the base be 1? 1 4 =1 1 10 =1 That would mean log 1 1=4 Log 1 1=10 That would be ambiguous, so we just don’t let it happen.

105
Why must the argument be > 0? 5 2 =25 and 25 is positive 5 0 =1 and 1 is positive 5 -2 = 1/25 and that’s positive too Since 5 to any power gives us a positive result, the argument has to be a positive number.

106
Find x in this example.

107
log x 36 = 2

108
log x 36 = 2 x 2 = 36

109
log x 36 = 2 6 2 = 36 (-6) 2 = 36

110
log x 36 = 2 x=6 6 2 = 36 (-6) 2 = 36 -6 would make the base < 0

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google