1. The Logarithmic Functions

2. The Logarithmic Functions
There are three numbers in an exponential notation.
4 3 = 64

3. The Logarithmic Functions
There are three numbers in an exponential notation.
4 3 = 64
the base

4. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
4 3 = 64
the base

5. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
the output
4 3 = 64
the base

6. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
the output
4 3 = 64
the base
Given the above expression, we say that
“(base) 4 raised to the exponent (power) 3 gives 64”.

7. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
the output
4 3 = 64
the base
Given the above expression, we say that
“(base) 4 raised to the exponent (power) 3 gives 64”.
The focus of the above statement is that when 43 is
executed, the output is 64.

8. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
the output
4 3 = 64
the base
Given the above expression, we say that
“(base) 4 raised to the exponent (power) 3 gives 64”.
The focus of the above statement is that when 43 is
executed, the output is 64.
However if we are given the output is 64 from
raising 4 to a power,

9. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
the output
4 3 = 64
the base
Given the above expression, we say that
“(base) 4 raised to the exponent (power) 3 gives 64”.
The focus of the above statement is that when 43 is
executed, the output is 64.
However if we are given the output is 64 from
raising 4 to a power,
the power
3
4 = 64
the base the output

10. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
the output
4 3 = 64
the base
Given the above expression, we say that
“(base) 4 raised to the exponent (power) 3 gives 64”.
The focus of the above statement is that when 43 is
executed, the output is 64.
However if we are given the output is 64 from
raising 4 to a power, then the
needed power is called the power = log4(64)
log4(64)
3
4 = 64
the base the output

11. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
the output
4 3 = 64
the base
Given the above expression, we say that
“(base) 4 raised to the exponent (power) 3 gives 64”.
The focus of the above statement is that when 43 is
executed, the output is 64.
However if we are given the output is 64 from
raising 4 to a power, then the
needed power is called the power = log4(64)
log4(64) which is 3.
3
4 = 64
the base the output

12. The Logarithmic Functions
There are three numbers in an exponential notation.
the exponent
the output
4 3 = 64
the base
Given the above expression, we say that
“(base) 4 raised to the exponent (power) 3 gives 64”.
The focus of the above statement is that when 43 is
executed, the output is 64.
However if we are given the output is 64 from
raising 4 to a power, then the
needed power is called the power = log4(64)
log4(64) which is 3.
3
or that log4(64) = 3 and we say 4 = 64
that “log–base–4 of 64 is 3”. the base the output

13. The Logarithmic Functions
Just as the sentence
“Bart's dad is Homer.”
contains the same information as
“Homer's son is Bart.”,

14. The Logarithmic Functions
Just as the sentence
“Bart's dad is Homer.”
contains the same information as
“Homer's son is Bart.” The expression
“64 = 43”
contains the same information as
“log4(64) = 3”.

15. The Logarithmic Functions
Just as the sentence
“Bart's dad is Homer.”
contains the same information as
“Homer's son is Bart.” The expression
“64 = 43”
contains the same information as
“log4(64) = 3”.
The expression “64 = 43” is called the exponential form
and “log4(64) = 3” is called the logarithmic form of the
expressed relation.

16. The Logarithmic Functions
Just as the sentence
“Bart's dad is Homer.”
contains the same information as
“Homer's son is Bart.” The expression
“64 = 43”
contains the same information as
“log4(64) = 3”.
The expression “64 = 43” is called the exponential form
and “log4(64) = 3” is called the logarithmic form of the
expressed relation.
In general, we say that
“log–base–b of y is x” or
logb(y) = x

17. The Logarithmic Functions
Just as the sentence
“Bart's dad is Homer.”
contains the same information as
“Homer's son is Bart.” The expression
“64 = 43”
contains the same information as
“log4(64) = 3”.
The expression “64 = 43” is called the exponential form
and “log4(64) = 3” is called the logarithmic form of the
expressed relation.
In general, we say that
“log–base–b of y is x” or
logb(y) = x if y = bx (b > 0).

18. The Logarithmic Functions
Just as the sentence
“Bart's dad is Homer.”
contains the same information as
“Homer's son is Bart.” The expression
“64 = 43”
contains the same information as
“log4(64) = 3”.
The expression “64 = 43” is called the exponential form
and “log4(64) = 3” is called the logarithmic form of the
expressed relation.
the power = logb(y)
In general, we say that
“log–base–b of y is x” or x
logb(y) = x if y = b x (b > 0). b =y
the base the output

19. The Logarithmic Functions
Just as the sentence
“Bart's dad is Homer.”
contains the same information as
“Homer's son is Bart.” The expression
“64 = 43”
contains the same information as
“log4(64) = 3”.
The expression “64 = 43” is called the exponential form
and “log4(64) = 3” is called the logarithmic form of the
expressed relation.
the power = logb(y)
In general, we say that
“log–base–b of y is x” or x
logb(y) = x if y = b x (b > 0), b =y
i.e. logb(y) is the exponent x. the base the output

20. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.

21. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.

22. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
exp–form
43 → 64
82 → 64
26 → 64

23. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64
by the bases and the
82 → 64
different exponents
required. 26 → 64

24. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64 log4(64)
by the bases and the
82 → 64 log8(64)
different exponents
required. 26 → 64 log2(64)

25. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64 log4(64) →
by the bases and the
82 → 64 log8(64) →
different exponents
required. 26 → 64 log2(64) →

26. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64 log4(64) → 3
by the bases and the
82 → 64 log8(64) →
different exponents
required. 26 → 64 log2(64) →

27. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64 log4(64) → 3
by the bases and the
82 → 64 log8(64) → 2
different exponents
required. 26 → 64 log2(64) →

28. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64 log4(64) → 3
by the bases and the
82 → 64 log8(64) → 2
different exponents
required. 26 → 64 log2(64) → 6

29. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64 log4(64) → 3
by the bases and the
82 → 64 log8(64) → 2
different exponents
required. 26 → 64 log2(64) → 6
Both numbers b and y appeared in the log notation
“logb(y)” must be positive.

30. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64 log4(64) → 3
by the bases and the
82 → 64 log8(64) → 2
different exponents
required. 26 → 64 log2(64) → 6
Both numbers b and y appeared in the log notation
“logb(y)” must be positive. Switch to x as the input,
the domain of logb(x) is the set D = {x l x > 0 }.

31. The Logarithmic Functions
When working with the exponential form or the
logarithmic expressions, always identify the base
number b first.
All the following exponential expressions yield 64.
Their corresponding exp–form log–form
log–form are differentiated 43 → 64 log4(64) → 3
by the bases and the
82 → 64 log8(64) → 2
different exponents
required. 26 → 64 log2(64) → 6
Both numbers b and y appeared in the log notation
“logb(y)” must be positive. Switch to x as the input,
the domain of logb(x) is the set D = {x l x > 0 }.
We would get an error message if we execute
log2(–1) with software.

32. The Logarithmic Functions
To convert the exp-form to the log–form:
x
b =y

33. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Identity the base and the
correct log–function

34. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
insert the exponential
output.

35. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
The log–output is the
required exponent.

36. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16
b. w = u2+v

37. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v

38. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v

39. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v

40. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v

41. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v

42. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v

43. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
logb( y ) = x

44. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y

45. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y

46. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y

47. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2
b. 2w = logv(a – b)

48. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2  3-2 = 1/9
b. 2w = logv(a – b)

49. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2  3-2 = 1/9
b. 2w = logv(a – b)

50. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2  3-2 = 1/9
b. 2w = logv(a – b)

51. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2  3-2 = 1/9
b. 2w = logv(a – b)  v2w = a – b

52. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2  3-2 = 1/9
b. 2w = logv(a – b)  v2w = a – b

53. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2  3-2 = 1/9
b. 2w = logv(a – b)  v2w = a – b

54. The Logarithmic Functions
To convert the exp-form to the log–form:
→
x
b =y logb( y ) = x
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16  log4(16) = 2
b. w = u2+v  logu(w) = 2+v
To convert the log–form to the exp–form:
→
x
logb( y ) = x b =y
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2  3-2 = 1/9
b. 2w = logv(a – b)  v2w = a – b
The output of logb(x), i.e. the exponent in the defined
relation, may be positive or negative.

55. The Logarithmic Functions
Example C.
a. Rewrite the exp-form into the log-form.
exp–form log–form
4–3 = 1/64 log4(1/64) = –3
8–2 = 1/64 log8(1/64) = –2
b. Rewrite the log-form into the exp-form.
log–form exp–form
log1/2(4) = –2 (1/2)–2 = 4
log1/2(8) = –3 (1/2)–3 = 8

56. The Logarithmic Functions
Example C.
a. Rewrite the exp-form into the log-form.
exp–form log–form
4–3 = 1/64 log4(1/64) = –3
8–2 = 1/64 log8(1/64) = –2
b. Rewrite the log-form into the exp-form.
log–form exp–form
log1/2(4) = –2 (1/2)–2 = 4
log1/2(8) = –3 (1/2)–3 = 8
The Common Log and the Natural Log

57. The Logarithmic Functions
Example C.
a. Rewrite the exp-form into the log-form.
exp–form log–form
4–3 = 1/64 log4(1/64) = –3
8–2 = 1/64 log8(1/64) = –2
b. Rewrite the log-form into the exp-form.
log–form exp–form
log1/2(4) = –2 (1/2)–2 = 4
log1/2(8) = –3 (1/2)–3 = 8
The Common Log and the Natural Log
Base 10 is called the common base.

58. The Logarithmic Functions
Example C.
a. Rewrite the exp-form into the log-form.
exp–form log–form
4–3 = 1/64 log4(1/64) = –3
8–2 = 1/64 log8(1/64) = –2
b. Rewrite the log-form into the exp-form.
log–form exp–form
log1/2(4) = –2 (1/2)–2 = 4
log1/2(8) = –3 (1/2)–3 = 8
The Common Log and the Natural Log
Base 10 is called the common base. Log with
base10,. written as log(x) without the base number b,
is called the common log,

59. The Logarithmic Functions
Example C.
a. Rewrite the exp-form into the log-form.
exp–form log–form
4–3 = 1/64 log4(1/64) = –3
8–2 = 1/64 log8(1/64) = –2
b. Rewrite the log-form into the exp-form.
log–form exp–form
log1/2(4) = –2 (1/2)–2 = 4
log1/2(8) = –3 (1/2)–3 = 8
The Common Log and the Natural Log
Base 10 is called the common base. Log with
base10,. written as log(x) without the base number b,
is called the common log, i.e. log(x) is log10(x).

60. The Common Log and the Natural Log
Base e is called the natural base.

61. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log,

62. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log, i.e. In(x) is loge(x).

63. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log, i.e. In(x) is loge(x).
Example D. Convert to the other form.
exp-form log-form
103 = 1000
ln(1/e2) = -2
e rt = A
P
log(1) = 0

64. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log, i.e. In(x) is loge(x).
Example D. Convert to the other form.
exp-form log-form
103 = 1000 log(1000) = 3
ln(1/e2) = -2
e rt = A
P
log(1) = 0

65. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log, i.e. In(x) is loge(x).
Example D. Convert to the other form.
exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
e rt = A
P
log(1) = 0

66. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log, i.e. In(x) is loge(x).
Example D. Convert to the other form.
exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
e rt = A ln( A ) = rt
P P
log(1) = 0

67. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log, i.e. In(x) is loge(x).
Example D. Convert to the other form.
exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
e rt = A ln( A ) = rt
P P
100 = 1 log(1) = 0

68. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log, i.e. In(x) is loge(x).
Example D. Convert to the other form.
exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
e rt = A ln( A ) = rt
P P
100 = 1 log(1) = 0
Most log and powers can’t be computed efficiently by
hand.

69. The Common Log and the Natural Log
Base e is called the natural base.
Log with base e is written as ln(x) and it’s called
the natural log, i.e. In(x) is loge(x).
Example D. Convert to the other form.
exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
e rt = A ln( A ) = rt
P P
100 = 1 log(1) = 0
Most log and powers can’t be computed efficiently by
hand. We need a calculation device to extract
numerical solutions.

70. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) =

71. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...

72. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 =

73. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50

74. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) =

75. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) = 2.1972245..

76. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) = 2.1972245..
In the exp–form, it’s e2.1972245 =

77. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) = 2.1972245..
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9

78. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) = 2.1972245..
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
c. Calculate the power using a calculator.
Then convert the relation into the log–form and
confirm the log–form by the calculator.
e4.3 =

79. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) = 2.1972245..
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
c. Calculate the power using a calculator.
Then convert the relation into the log–form and
confirm the log–form by the calculator.
e4.3 = 73.699793..

80. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) = 2.1972245..
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
c. Calculate the power using a calculator.
Then convert the relation into the log–form and
confirm the log–form by the calculator.
e4.3 = 73.699793..→ In(73.699793) =

81. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) = 2.1972245..
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
c. Calculate the power using a calculator.
Then convert the relation into the log–form and
confirm the log–form by the calculator.
e4.3 = 73.699793..→ In(73.699793) = 4.299999..≈ 4.3

82. The Common Log and the Natural Log
Example E. Calculate each of the following logs
using a calculator. Then convert the relation into the
exp–form and confirm the exp–form with a calculator.
a. log(50) = 1.69897...
In the exp–form, it’s101.69897 = 49.9999995...≈50
b. ln(9) = 2.1972245..
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
c. Calculate the power using a calculator.
Then convert the relation into the log–form and
confirm the log–form by the calculator.
e4.3 = 73.699793..→ In(73.699793) = 4.299999..≈ 4.3
Your turn. Follow the instructions in part c for 10π.

83. The Common Log and the Natural Log
Equation may be formed with log–notation.

84. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form.

85. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.

86. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1

87. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.

88. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.
So x = 1/9

89. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.
So x = 1/9
b. logx(9) = –2

90. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.
So x = 1/9
b. logx(9) = –2
Drop the log and get 9 = x–2,

91. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.
So x = 1/9
b. logx(9) = –2
1
Drop the log and get 9 = x–2, i.e. 9 = x2

92. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.
So x = 1/9
b. logx(9) = –2
1
Drop the log and get 9 = x–2, i.e. 9 = x2
So 9x2 = 1

93. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.
So x = 1/9
b. logx(9) = –2
1
Drop the log and get 9 = x–2, i.e. 9 = x2
So 9x2 = 1
x2 = 1/9
x = 1/3 or x= –1/3

94. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we
need to restate them in the exp–form. We say we
"drop the log" when this step is taken.
Example F. Solve for x
a. log9(x) = –1
Drop the log and get x = 9–1.
So x = 1/9
b. logx(9) = –2
1
Drop the log and get 9 = x–2, i.e. 9 = x2
So 9x2 = 1
x2 = 1/9
x = 1/3 or x= –1/3
Since the base b > 0, so x = 1/3 is the only solution.

95. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Graphs of the Logarithmic Functions

96. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s.
Graphs of the Logarithmic Functions
x y=log2(x)
1/4
1/2
1
2
4
8

97. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
Graphs of the Logarithmic Functions
x y=log2(x)
2
4
8

98. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
Graphs of the Logarithmic Functions
x y=log2(x)
1/4
1/2
1
2
4
8

99. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
Graphs of the Logarithmic Functions
x y=log2(x)
1/4 -2
1/2
1
2
4
8

100. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
Graphs of the Logarithmic Functions
x y=log2(x)
1/4 -2
1/2 -1
1
2
4
8

101. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
Graphs of the Logarithmic Functions
x y=log2(x)
1/4 -2
1/2 -1
1 0
2
4
8

102. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
Graphs of the Logarithmic Functions
x y=log2(x)
1/4 -2
1/2 -1
1 0
2 1
4
8

103. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
Graphs of the Logarithmic Functions
x y=log2(x)
1/4 -2
1/2 -1
1 0
2 1
4 2
8 3

104. The Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
Hence to make a table to plot the graph of y = log2(x),
we only select positive x’s. In particular we select x’s
related to base 2 for easy computation of the y’s.
Graphs of the Logarithmic Functions
x y=log2(x) y
1/4 -2 (16, 4)
(8, 3)
1/2 -1 (4, 2)
(2, 1)
1 0 x
(1, 0)
2 1 (1/2, -1)
4 2 (1/4, -2)
y=log2(x)
8 3

105. The Logarithmic Functions
To graph log with base b = ½, we have
log1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4

106. The Logarithmic Functions
To graph log with base b = ½, we have
log1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4
y
y = log1/2(x)
(1, 0) x
(4, -2)
(8, -3)
(16, -4)

107. The Logarithmic Functions
To graph log with base b = ½, we have
log1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4
y
y = log1/2(x)
(1, 0) x
(4, -2)
(8, -3)
(16, -4)
Here are the general shapes of log–functions.
y y y = logb(x), 1 > b
(b, 1)
(b, 1) x x
(1, 0) (1, 0)
y = logb(x), b > 1