1. The Logarithmic Functions

2. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10x
that the output y encompasses all
the positive numbers.
x

3. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10x
that the output y encompasses all
the positive numbers.
x

4. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10x
that the output y encompasses all
the positive numbers.
x
In fact, given any positive number y,
there is exactly one x such that 10x = y.

5. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10x
that the output y encompasses all
the positive numbers.
x
In fact, given any positive number y,
there is exactly one x such that 10x = y.
For example, if y is 100,

6. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10x
that the output y encompasses all
the positive numbers.
x
In fact, given any positive number y,
there is exactly one x such that 10x = y.
For example, if y is 100, since100 = 102
then x must be 2,

7. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10
x
that the output y encompasses all
the positive numbers.
x
In fact, given any positive number y,
there is exactly one x such that 10x = y.
For example, if y is 100, since100 = 102
then x must be 2, i.e. log(100) = 2.
Similarly if y is 5, then x must be log(5) = 0.6989..
since 5 = 100.6989...

8. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10
x
that the output y encompasses all
the positive numbers.
x
In fact, given any positive number y,
there is exactly one x such that 10x = y.
For example, if y is 100, since100 = 102
then x must be 2, i.e. log(100) = 2.
Similarly if y is 5, then x must be log(5) = 0.6989..
since 5 = 100.6989... So log(y) is a well defined function.

9. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10
x
that the output y encompasses all
the positive numbers.
x
In fact, given any positive number y,
there is exactly one x such that 10x = y.
For example, if y is 100, since100 = 102
then x must be 2, i.e. log(100) = 2.
Similarly if y is 5, then x must be log(5) = 0.6989..
since 5 = 100.6989... So log(y) is a well defined function.
This is also the case for all other positive bases b .

10. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10
x
that the output y encompasses all
the positive numbers.
x
In fact, given any positive number y,
there is exactly one x such that 10x = y.
For example, if y is 100, since100 = 102
then x must be 2, i.e. log(100) = 2.
Similarly if y is 5, then x must be log(5) = 0.6989..
since 5 = 100.6989... So log(y) is a well defined function.
This is also the case for all other positive bases b .
The Existence of logb( y )

11. The Logarithmic Functions
y
From the graph of y = 10 we see
x y = 10
x
that the output y encompasses all
the positive numbers.
x
In fact, given any positive number y,
there is exactly one x such that 10x = y.
For example, if y is 100, since100 = 102
then x must be 2, i.e. log(100) = 2.
Similarly if y is 5, then x must be log(5) = 0.6989..
since 5 = 100.6989... So log(y) is a well defined function.
This is also the case for all other positive bases b .
The Existence of logb( y )
Given any positive base b (b ≠ 1) and any positive
number y, there is exactly one x such that y = bx
i.e. x = logb(y) is well defined.

12. Properties of Logarithm

13. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:

14. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1

15. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0

16. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t

17. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)

18. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
In this version,
logb(x) corresponds to r,
logb(y) corresponds to t.

19. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
br = br-t
3. t
b

20. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
br = br-t
3. t x
b 3. logb( y ) = logb(x) – logb(y)

21. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
3. tbr = br-t x
b 3. logb( y ) = logb(x) – logb(y)
4. (br)t = brt

22. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
3. tbr = br-t x
b 3. logb( y ) = logb(x) – logb(y)
4. (br)t = brt
4. logb(xt) = t·logb(x)

23. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
3. tbr = br-t x
b 3. logb( y ) = logb(x) – logb(y)
4. (br)t = brt
4. logb(xt) = t·logb(x)
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:

24. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
3. tbr = br-t x
b 3. logb( y ) = logb(x) – logb(y)
4. (br)t = brt
4. logb(xt) = t·logb(x)
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers.

25. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
3. tbr = br-t x
b 3. logb( y ) = logb(x) – logb(y)
4. (br)t = brt
4. logb(xt) = t·logb(x)
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers. Let log b(x) = r
and logb(y) = t, which in exp-form are x = br and y = bt.

26. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
3. tbr = br-t x
b 3. logb( y ) = logb(x) – logb(y)
4. (br)t = brt
4. logb(xt) = t·logb(x)
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers. Let log b(x) = r
and logb(y) = t, which in exp-form are x = br and y = bt.
Therefore x·y = br+t,

27. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
3. tbr = br-t x
b 3. logb( y ) = logb(x) – logb(y)
4. (br)t = brt
4. logb(xt) = t·logb(x)
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers. Let log b(x) = r
and logb(y) = t, which in exp-form are x = br and y = bt.
Therefore x·y = br+t, which in log-form is
logb(x·y) = r + t = logb(x)+logb(y).

28. Properties of Logarithm
Recall the following The corresponding
Rules of Exponents: Rules of Logs are:
1. b0 = 1 1. logb(1) = 0
2. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
3. tbr = br-t x
b 3. logb( y ) = logb(x) – logb(y)
4. (br)t = brt
4. logb(xt) = t·logb(x)
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers. Let log b(x) = r
and logb(y) = t, which in exp-form are x = br and y = bt.
Therefore x·y = br+t, which in log-form is
The(x·y) = rules = logbbe verified similarly.
logb other r + t may (x)+logb(y).

29. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y

30. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2 ),
√y y1/2

31. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2 ), by the quotient rule
√y y1/2
= log (3x2) – log(y1/2)

32. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2 ), by the quotient rule
√y y1/2
= log (3x2) – log(y1/2)
product rule
= log(3) + log(x2)

33. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2), by the quotient rule
√y y1/2
= log (3x2) – log(y1/2)
product rule power rule
= log(3) + log(x2) – ½ log(y)

34. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2), by the quotient rule
√y y1/2
= log (3x2) – log(y1/2)
product rule power rule
= log(3) + log(x2) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)

35. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2), by the quotient rule
√y y1/2
= log (3x2) – log(y1/2)
product rule power rule
= log(3) + log(x2) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
b. Combine log(3) + 2log(x) – ½ log(y) into one log.

36. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2), by the quotient rule
√y y1/2
= log (3x2) – log(y1/2)
product rule power rule
= log(3) + log(x2) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
b. Combine log(3) + 2log(x) – ½ log(y) into one log.
log(3) + 2log(x) – ½ log(y) power rule
= log(3) + log(x2) – log(y1/2)

37. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2), by the quotient rule
√y y1/2
= log (3x2) – log(y1/2)
product rule power rule
= log(3) + log(x2) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
b. Combine log(3) + 2log(x) – ½ log(y) into one log.
log(3) + 2log(x) – ½ log(y) power rule
= log(3) + log(x2) – log(y1/2) product rule
= log (3x2) – log(y1/2)

38. Properties of Logarithm
Example A.
a. Write log( 3x2 ) in terms of log(x) and log(y).
√y
log( 3x2 ) = log( 3x2), by the quotient rule
√y y1/2
= log (3x2) – log(y1/2)
product rule power rule
= log(3) + log(x2) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
b. Combine log(3) + 2log(x) – ½ log(y) into one log.
log(3) + 2log(x) – ½ log(y) power rule
= log(3) + log(x2) – log(y1/2) product rule
2 1/2 3x2)
= log (3x ) – log(y )= log( 1/2
y

39. Properties of Logarithm
The exponential function bx is also written as expb(x).

40. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.

41. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines.

42. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other.

43. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.

44. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2

45. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10

46. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100

47. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100 log (100)
10

48. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100 log (100)
10
2 (the starting x)

49. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100 log (100)
10
2 (the starting x)
x = 100

50. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100 log (100)
10
2 (the starting x)
x = 100 log10(100)

51. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100 log (100)
10
2 (the starting x)
x = 100 log10(100) 2

52. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100 log (100)
10
2 (the starting x)
x = 100 log10(100) 2 exp10(2)

53. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100 log (100)
10
2 (the starting x)
x = 100 log10(100) 2 exp10(2) 100 (the starting x)

54. Properties of Logarithm
The exponential function bx is also written as expb(x).
For example, exp10(x) is 10x and exp10(2) = 102 = 100.
The pair of functions expb(x) and logb(x) scramble and
unscramble the output of each other like a pair of
coding–decoding machines. An input x after being
processed by one function may be de–processed by
the other. We will illustrate this with the pair exp10(x),
log (x) and the relation 102 = 100.
Starting with an input, say
x=2 exp (2)
10
100 log (100)
10
2 (the starting x)
x = 100 log10(100) 2 exp10(2) 100 (the starting x)
A pair of functions such as expb(x) and logb(x) that
unscramble each other is called an inverse pair.

55. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x

56. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) #

57. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x

58. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
b

59. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
a. logb(expb(x)) = x or logb(bx) = x
b

60. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
x (x > 0) logb (x) # expb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
a. logb(expb(x)) = x or logb(bx) = x
b
b. expb(logb(x)) = x or blog (x) = x

61. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
x (x > 0) logb (x) # expb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
a. logb(expb(x)) = x or logb(bx) = x
b
b. expb(logb(x)) = x or blog (x) = x
Example B: Simplify
a. log2(2–5) =
b. 8log 8(xy) =
c. e2 + ln(7) =

62. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
x (x > 0) logb (x) # expb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
a. logb(expb(x)) = x or logb(bx) = x
b
b. expb(logb(x)) = x or blog (x) = x
Example B: Simplify
a. log2(2–5) = –5
b. 8log 8(xy) =
c. e2 + ln(7) =

63. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
x (x > 0) logb (x) # expb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
a. logb(expb(x)) = x or logb(bx) = x
b
b. expb(logb(x)) = x or blog (x) = x
Example B: Simplify
a. log2(2–5) = –5
b. 8log 8(xy) = xy
c. e2 + ln(7) =

64. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
x (x > 0) logb (x) # expb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
a. logb(expb(x)) = x or logb(bx) = x
b
b. expb(logb(x)) = x or blog (x) = x
Example B: Simplify
a. log2(2–5) = –5
b. 8log 8(xy) = xy
c. e2 + ln(7) = e2·eln(7)

65. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
x (x > 0) logb (x) # expb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
a. logb(expb(x)) = x or logb(bx) = x
b
b. expb(logb(x)) = x or blog (x) = x
Example B: Simplify
a. log2(2–5) = –5
b. 8log 8(xy) = xy
c. e2 + ln(7) = e2·eln(7)

66. Properties of Logarithm
So for all pairs of expb(x) & logb(x) and an input x
x expb (x) # logb(#) x
x (x > 0) logb (x) # expb(#) x
Here is the inverse relation stated in function notation.
The Inverse Relation of Exp and Log
a. logb(expb(x)) = x or logb(bx) = x
b
b. expb(logb(x)) = x or blog (x) = x
Example B: Simplify
a. log2(2–5) = –5
b. 8log 8(xy) = xy
c. e2 + ln(7) = e2·eln(7) = 7e2