64 introduction to logarithm
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The Logarithmic Functions
There are three numbers in an exponential notation.The Logarithmic Functions
4 3 = 64
There are three numbers in an exponential notation.The Logarithmic Functions
the base4 3 = 64
There are three numbers in an exponential notation.The Logarithmic Functions
the exponent
the base4 3 = 64
There are three numbers in an exponential notation.The Logarithmic Functions
the exponent
the base
the output
4 3 = 64
There are three numbers in an exponential notation.
Given the above expression, we say that “(base) 4 raised to the exponent (power) 3 gives 64”.
The Logarithmic Functions
the exponent
the base
the output
4 3 = 64
There are three numbers in an exponential notation.The Logarithmic Functions
the exponent
the base
the output
4 3 = 64
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.
There are three numbers in an exponential notation.The Logarithmic Functions
the exponent
the base
the output
4 3 = 64
However if we are given the output is 64 from raising 4 to a power,
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.
There are three numbers in an exponential notation.The Logarithmic Functions
the exponent
the base
the output
4 3 = 64
However if we are given the output is 64 from raising 4 to a power,
the power
the base the output
4 = 643
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.
There are three numbers in an exponential notation.The Logarithmic Functions
the exponent
the base
the output
4 3 = 64
However if we are given the output is 64 from raising 4 to a power, then the needed power is called log4(64)
the power = log4(64)
the base the output
4 = 643
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.
There are three numbers in an exponential notation.The Logarithmic Functions
the exponent
the base
the output
4 3 = 64
However if we are given the output is 64 from raising 4 to a power, then the needed power is called log4(64) which is 3.
the power = log4(64)
the base the output
4 = 643
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.
There are three numbers in an exponential notation.The Logarithmic Functions
the exponent
the base
the output
4 3 = 64
However if we are given the output is 64 from raising 4 to a power, then the needed power is called log4(64) which is 3.
the power = log4(64)
the base the output
4 = 643
or that log4(64) = 3 and we say that “log–base–4 of 64 is 3”.
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.
The Logarithmic FunctionsJust as the sentence “Bart's dad is Homer.” contains the same information as “Homer's son is Bart.”,
The Logarithmic FunctionsJust 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 Logarithmic FunctionsJust 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 Logarithmic FunctionsJust 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
The Logarithmic FunctionsJust 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).
The Logarithmic FunctionsJust 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).
the power = logb(y)
the base the output
b = yx
The Logarithmic FunctionsJust 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),i.e. logb(y) is the exponent x.
the power = logb(y)
the base the output
b = yx
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first.
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
43 → 64
82 → 64
26 → 64
exp–form
The Logarithmic FunctionsWhen 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 log–form are differentiated by the bases and the different exponents required.
43 → 64
82 → 64
26 → 64
exp–form log–form
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
43 → 64
82 → 64
26 → 64
log4(64)
log8(64)
log2(64)
exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
The Logarithmic FunctionsWhen 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 log–form are differentiated by the bases and the different exponents required.
43 → 64
82 → 64
26 → 64
log4(64) →
log8(64) →
log2(64) →
exp–form log–form
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
43 → 64
82 → 64
26 → 64
log4(64) → 3
log8(64) →
log2(64) →
exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
43 → 64
82 → 64
26 → 64
log4(64) → 3
log8(64) → 2
log2(64) →
exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
43 → 64
82 → 64
26 → 64
log4(64) → 3
log8(64) → 2
log2(64) → 6
exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
43 → 64
82 → 64
26 → 64
log4(64) → 3
log8(64) → 2
log2(64) → 6
exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
Both numbers b and y appeared in the log notation “logb(y)” must be positive.
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
43 → 64
82 → 64
26 → 64
log4(64) → 3
log8(64) → 2
log2(64) → 6
exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
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 }.
The Logarithmic FunctionsWhen working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
43 → 64
82 → 64
26 → 64
log4(64) → 3
log8(64) → 2
log2(64) → 6
exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
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.
The Logarithmic FunctionsTo convert the exp-form to the log–form:
b = yx
The Logarithmic FunctionsTo convert the exp-form to the log–form:
b = yx
logb( y ) = x→Identity the base and the correct log–function
The Logarithmic FunctionsTo convert the exp-form to the log–form:
b = yx
logb( y ) = x→insert the exponential output.
The Logarithmic FunctionsTo convert the exp-form to the log–form:
b = yx
logb( y ) = x→The log–output is the required exponent.
The Logarithmic Functions
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16 b. w = u2+v
To convert the exp-form to the log–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16 log4(16) = 2 b. w = u2+v
To convert the exp-form to the log–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16 log4(16) = 2 b. w = u2+v
To convert the exp-form to the log–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example A. Rewrite the exp-form into the log-form.
a. 42 = 16 log4(16) = 2 b. w = u2+v
To convert the exp-form to the log–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
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 exp-form to the log–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
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 exp-form to the log–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
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 exp-form to the log–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
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 exp-form to the log–form:
b = yx
→
To convert the log–form to the exp–form:
logb( y ) = x
logb( y ) = x
The Logarithmic Functions
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 exp-form to the log–form:
b = yx
→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
logb( y ) = x
The Logarithmic Functions
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 exp-form to the log–form:
b = yx
→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
logb( y ) = x
The Logarithmic Functions
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 exp-form to the log–form:
b = yx
→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
logb( y ) = x
The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2b. 2w = logv(a – b)
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 exp-form to the log–form:
b = yx
logb( y ) = x→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2 3-2 = 1/9b. 2w = logv(a – b)
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 exp-form to the log–form:
b = yx
logb( y ) = x→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2 3-2 = 1/9b. 2w = logv(a – b)
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 exp-form to the log–form:
b = yx
logb( y ) = x→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2 3-2 = 1/9b. 2w = logv(a – b)
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 exp-form to the log–form:
b = yx
logb( y ) = x→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2 3-2 = 1/9b. 2w = logv(a – b) v2w = a – b
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 exp-form to the log–form:
b = yx
logb( y ) = x→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2 3-2 = 1/9b. 2w = logv(a – b) v2w = a – b
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 exp-form to the log–form:
b = yx
logb( y ) = x→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2 3-2 = 1/9b. 2w = logv(a – b) v2w = a – b
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 exp-form to the log–form:
b = yx
logb( y ) = x→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
The Logarithmic Functions
Example B. Rewrite the log-form into the exp-form.
a. log3(1/9) = –2 3-2 = 1/9b. 2w = logv(a – b) v2w = a – b
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 exp-form to the log–form:
b = yx
logb( y ) = x→
To convert the log–form to the exp–form:
b = yx
logb( y ) = x→
The output of logb(x), i.e. the exponent in the defined relation, may be positive or negative.
The Logarithmic FunctionsExample C. a. Rewrite the exp-form into the log-form.
4–3 = 1/64
8–2 = 1/64
log4(1/64) = –3
log8(1/64) = –2
exp–form log–form
b. Rewrite the log-form into the exp-form.
(1/2)–2 = 4log1/2(4) = –2
log1/2(8) = –3
exp–formlog–form
(1/2)–3 = 8
The Logarithmic Functions
The Common Log and the Natural Log
Example C. a. Rewrite the exp-form into the log-form.
4–3 = 1/64
8–2 = 1/64
log4(1/64) = –3
log8(1/64) = –2
exp–form log–form
b. Rewrite the log-form into the exp-form.
(1/2)–2 = 4log1/2(4) = –2
log1/2(8) = –3
exp–formlog–form
(1/2)–3 = 8
The Logarithmic Functions
Base 10 is called the common base.The Common Log and the Natural Log
Example C. a. Rewrite the exp-form into the log-form.
4–3 = 1/64
8–2 = 1/64
log4(1/64) = –3
log8(1/64) = –2
exp–form log–form
b. Rewrite the log-form into the exp-form.
(1/2)–2 = 4log1/2(4) = –2
log1/2(8) = –3
exp–formlog–form
(1/2)–3 = 8
The Logarithmic Functions
Base 10 is called the common base. Log with base10,. written as log(x) without the base number b, is called the common log,
The Common Log and the Natural Log
Example C. a. Rewrite the exp-form into the log-form.
4–3 = 1/64
8–2 = 1/64
log4(1/64) = –3
log8(1/64) = –2
exp–form log–form
b. Rewrite the log-form into the exp-form.
(1/2)–2 = 4log1/2(4) = –2
log1/2(8) = –3
exp–formlog–form
(1/2)–3 = 8
The Logarithmic Functions
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).
The Common Log and the Natural Log
Example C. a. Rewrite the exp-form into the log-form.
4–3 = 1/64
8–2 = 1/64
log4(1/64) = –3
log8(1/64) = –2
exp–form log–form
b. Rewrite the log-form into the exp-form.
(1/2)–2 = 4log1/2(4) = –2
log1/2(8) = –3
exp–formlog–form
(1/2)–3 = 8
Base e is called the natural base. 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,
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).
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).
The Common Log and the Natural Log
Example D. Convert to the other form. exp-form log-form
103 = 1000
ln(1/e2) = -2
ert =
log(1) = 0
AP
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).
The Common Log and the Natural Log
Example D. Convert to the other form. exp-form log-form
103 = 1000 log(1000) = 3
ln(1/e2) = -2
ert =
log(1) = 0
AP
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).
The Common Log and the Natural Log
Example D. Convert to the other form. exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
ert =
log(1) = 0
AP
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).
The Common Log and the Natural Log
Example D. Convert to the other form. exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
ert = ln( ) = rt
log(1) = 0
AP
AP
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).
The Common Log and the Natural Log
Example D. Convert to the other form. exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
ert = ln( ) = rt
100 = 1 log(1) = 0
AP
AP
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).
The Common Log and the Natural Log
Example D. Convert to the other form. exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
ert = ln( ) = rt
100 = 1 log(1) = 0
AP
AP
Most log and powers can’t be computed efficiently by hand.
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).
The Common Log and the Natural Log
Example D. Convert to the other form. exp-form log-form
103 = 1000 log(1000) = 3
e-2 = 1/e2 ln(1/e2) = -2
ert = ln( ) = rt
100 = 1 log(1) = 0
AP
AP
Most log and powers can’t be computed efficiently by hand. We need a calculation device to extract numerical solutions.
The Common Log and the Natural LogExample 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) =
The Common Log and the Natural LogExample 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...
The Common Log and the Natural LogExample 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 =
The Common Log and the Natural LogExample 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
The Common Log and the Natural LogExample 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) =
The Common Log and the Natural LogExample 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..
The Common Log and the Natural LogExample 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 =
The Common Log and the Natural LogExample 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
The Common Log and the Natural LogExample 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..
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 =
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
The Common Log and the Natural LogExample 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..
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 the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
The Common Log and the Natural LogExample 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..
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) =
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
The Common Log and the Natural LogExample 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..
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
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
The Common Log and the Natural LogExample 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..
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.
In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
Equation may be formed with log–notation. The Common Log and the Natural Log
Equation may be formed with log–notation. Often we need to restate them in the exp–form.
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1Drop the log and get x = 9–1.
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1Drop the log and get x = 9–1. So x = 1/9
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1Drop the log and get x = 9–1. So x = 1/9
b. logx(9) = –2
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1Drop the log and get x = 9–1. So x = 1/9
b. logx(9) = –2
Drop the log and get 9 = x–2,
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1Drop the log and get x = 9–1. So x = 1/9
b. logx(9) = –2
Drop the log and get 9 = x–2, i.e. 9 = 1x2
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1Drop the log and get x = 9–1. So x = 1/9
b. logx(9) = –2
Drop the log and get 9 = x–2, i.e. 9 = So 9x2 = 1
1x2
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1Drop the log and get x = 9–1. So x = 1/9
b. logx(9) = –2
Drop the log and get 9 = x–2, i.e. 9 = So 9x2 = 1 x2 = 1/9 x = 1/3 or x= –1/3
1x2
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.
The Common Log and the Natural Log
Example F. Solve for x
a. log9(x) = –1Drop the log and get x = 9–1. So x = 1/9
b. logx(9) = –2
Drop the log and get 9 = x–2, i.e. 9 = So 9x2 = 1 x2 = 1/9 x = 1/3 or x= –1/3Since the base b > 0, so x = 1/3 is the only solution.
1x2
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.
The Common Log and the Natural Log
The Logarithmic Functions
Graphs of the Logarithmic Functions
Recall that the domain of logb(x) is the set of all x > 0.
The Logarithmic Functions
Graphs of the Logarithmic Functions
1/4
1/2
1
2
4
8
x y=log2(x)
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.
The Logarithmic Functions
Graphs of the Logarithmic Functions
2
4
8
x y=log2(x)
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.
The Logarithmic Functions
Graphs of the Logarithmic Functions
1/4
1/2
1
2
4
8
x y=log2(x)
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.
The Logarithmic Functions
Graphs of the Logarithmic Functions
1/4 -21/2
1
2
4
8
x y=log2(x)
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.
The Logarithmic Functions
Graphs of the Logarithmic Functions
1/4 -21/2 -1
1
2
4
8
x y=log2(x)
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.
The Logarithmic Functions
Graphs of the Logarithmic Functions
1/4 -21/2 -1
1 0
2
4
8
x y=log2(x)
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.
The Logarithmic Functions
Graphs of the Logarithmic Functions
1/4 -21/2 -1
1 0
2 1
4
8
x y=log2(x)
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.
The Logarithmic Functions
Graphs of the Logarithmic Functions
1/4 -21/2 -1
1 0
2 1
4 2
8 3
x y=log2(x)
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.
The Logarithmic Functions
(1, 0)
(2, 1)(4, 2)
(8, 3)(16, 4)
(1/2, -1)
(1/4, -2)y=log2(x)
Graphs of the Logarithmic Functions
1/4 -21/2 -1
1 0
2 1
4 2
8 3
x y=log2(x)
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.
No G
raph Zone
x
y
The Logarithmic FunctionsTo graph log with base b = ½, we havelog1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4
The Logarithmic Functions
x
y
(1, 0)
(8, -3)
To graph log with base b = ½, we havelog1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4
(4, -2)
(16, -4)
y = log1/2(x)
No G
raph Zone
The Logarithmic Functions
x
y
(1, 0)
(8, -3)
To graph log with base b = ½, we havelog1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4
(4, -2)
(16, -4)
y = log1/2(x)
x x
y
(1, 0)(1, 0)
y = logb(x), b > 1
y = logb(x), 1 > b
Here are the general shapes of log–functions. yN
o Graph Z
one
No G
raph Zone
No G
raph Zone
(b, 1)(b, 1)
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