Log 10 X 2 Log 2x 10.
Daftar Isi:
- 1 Logarithm Rules
- 2 Logarithm definition
- 3 Logarithm as inverse function of exponential function
- 4 Natural logarithm (ln)
- 5 Inverse logarithm adding
- 6 Logarithm rules
- 6.0.0.1 Logarithm production rule
- 6.0.0.2 Logarithm caliber rule
- 6.0.0.3 Logarithm power dominion
- 6.0.0.4 Logarithm base of operations switch dominion
- 6.0.0.5 Logarithm base change rule
- 6.0.0.6 Derivative of logarithm
- 6.0.0.7 Integral of logarithm
- 6.0.0.8 Logarithm of negative number
- 6.0.0.9 Logarithm of 0
- 6.0.0.10 Logarithm of 1
- 6.0.0.11 Logarithm of the base
- 6.0.0.12 Logarithm of infinity
- 6.0.1 Logarithm product dominion
- 6.0.2 Logarithm caliber rule
- 6.0.3 Logarithm power rule
- 6.0.4 Logarithm base of operations switch dominion
- 6.0.5 Logarithm base change rule
- 6.0.6 Logarithm of negative number
- 6.0.7 Logarithm of 0
- 6.0.8 Logarithm of 1
- 6.0.9 Logarithm of infinity
- 6.0.10 Logarithm of the base
- 6.0.11 Logarithm derivative
- 6.0.12 Logarithm integral
- 7 Logarithm approximation
- 8 Complex logarithm
- 9 Logarithm bug and answers
- 10 Graph of log(x)
- 11 Logarithms table
- 12 See besides
- 13 Write how to improve this page
- 14 Log 10 X 2 Log 2x 10
Logarithm Rules
The
base
b
logarithm
of a number is the
exponent
that we need to raise the
base
in order to get the number.
- Logarithm definition
- Logarithm rules
- Logarithm problems
- Complex logarithm
- Graph of log(ten)
- Logarithm tabular array
- Logarithm estimator
Logarithm definition
When b is raised to the ability of y is equal x:
b^{
y}
=
x
Then the base of operations b logarithm of x is equal to y:
log
_{b}
(x)
= y
For example when:
2^{4}
= 16
Then
log_{two}(sixteen) = 4
Logarithm as inverse function of exponential function
The logarithmic office,
y
= log
_{b}
(x)
is the inverse function of the exponential role,
x
=
b^{y}
So if nosotros calculate the exponential function of the logarithm of x (x>0),
f
(f
^{-1}(x)) =
b
^{log}
b
^{(x)}
=
x
Or if we summate the logarithm of the exponential function of x,
f
^{-one}(f
(ten)) = log_{
b
}(b^{ten}
) =
x
Natural logarithm (ln)
Natural logarithm is a logarithm to the base e:
ln(x) = log
_{e}
(10)
When e constant is the number:
or
Come across: Natural logarithm
Inverse logarithm adding
The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y:
x
= log^{-1}(y) =
b^{
y}
Logarithmic function
The logarithmic function has the basic form of:
f
(x) = log
_{b}
(x)
Logarithm rules
Rule name | Rule |
---|---|
Logarithm production rule |
log _{b} (ten ∙ y) = log _{b} (10) + log _{b} (y) |
Logarithm caliber rule |
log _{b} (x / y) = log _{b} (x) – log _{b} (y) |
Logarithm power dominion |
log _{b} (10 ^{y} ) = y ∙ log _{b} (x) |
Logarithm base of operations switch dominion |
log _{b} (c) = i / log _{c} (b) |
Logarithm base change rule |
log _{b} (x) = log _{c} (10) / log _{c} (b) |
Derivative of logarithm |
f (ten) = log_{ b }(10) ⇒ f ‘ (x) = 1 / ( x ln(b) ) |
Integral of logarithm |
∫ log _{b} (x) dx = 10 ∙ ( log _{b} (x) – 1 / ln(b) ) + C |
Logarithm of negative number |
log_{
b
}(x) is undefined when x≤ 0 |
Logarithm of 0 |
log_{
b
}(0) is undefined |
Logarithm of 1 |
log_{ b }(ane) = 0 |
Logarithm of the base |
log_{ b }(b) = 1 |
Logarithm of infinity |
lim log_{
b
}(x) = ∞, when 10→∞ |
See: Logarithm rules
Logarithm product dominion
The logarithm of the multiplication of 10 and y is the sum of logarithm of x and logarithm of y.
log
_{b}
(x ∙ y) = log
_{b}
(x)
+
log
_{b}
(y)
For case:
log_{10}(3
∙
7) = log_{x}(3)
+
log_{x}(7)
Logarithm caliber rule
The logarithm of the segmentation of x and y is the difference of logarithm of 10 and logarithm of y.
log
_{b}
(x / y) = log
_{b}
(10)
–
log
_{b}
(y)
For example:
log_{10}(three
/
7) = log_{10}(three)
–
log_{10}(7)
Logarithm power rule
The logarithm of x raised to the ability of y is y times the logarithm of 10.
log
_{b}
(x
^{y}
) =
y ∙
log
_{b}
(x)
For case:
log_{10}(2^{
8
}) = eight∙
log_{10}(two)
Logarithm base of operations switch dominion
The base b logarithm of c is 1 divided by the base c logarithm of b.
log
_{b}
(c) = 1 / log
_{c}
(b)
For instance:
log_{ii}(8) = 1 / log_{8}(2)
Logarithm base change rule
The base b logarithm of 10 is base c logarithm of ten divided by the base c logarithm of b.
log
_{b}
(x) = log
_{c}
(x) / log
_{c}
(b)
For example, in order to calculate log_{2}(eight) in calculator, we need to change the base of operations to 10:
log_{2}(viii) = log_{x}(eight) / log_{10}(ii)
See: log base of operations change rule
Logarithm of negative number
The base of operations b real logarithm of x when ten<=0 is undefined when x is negative or equal to zero:
log_{
b
}(x)
is undefined when
x
≤ 0
Run across: log of negative number
Logarithm of 0
The base of operations b logarithm of zero is undefined:
log_{
b
}(0)
is undefined
The limit of the base of operations b logarithm of x, when x approaches zero, is minus infinity:
Encounter: log of cipher
Logarithm of 1
The base b logarithm of ane is nada:
log_{ b }(one) = 0
For example, teh base two logarithm of 1 is zero:
log_{2}(1) = 0
Meet: log of 1
Logarithm of infinity
The limit of the base b logarithm of ten, when x approaches infinity, is equal to infinity:
lim log_{
b
}(x) = ∞,
when
x→∞
See: log of infinity
Logarithm of the base
The base b logarithm of b is one:
log_{ b }(b) = 1
For example, the base two logarithm of two is 1:
log_{ii}(2) = 1
Logarithm derivative
When
f
(10) = log
_{b}
(x)
Then the derivative of f(x):
f ‘
(10) = 1 / (
x
ln(b) )
See: log derivative
Logarithm integral
The integral of logarithm of x:
∫
log
_{b}
(ten)
dx
=
x ∙
( log
_{b}
(ten)
– 1 / ln(b)
) +
C
For example:
∫
log_{2}(x)
dx
=
x ∙
( log_{2}(x)
– ane / ln(2)
) +
C
Logarithm approximation
log_{2}(x) ≈
due north
+ (10/2^{
north
}
– 1) ,
Complex logarithm
For circuitous number z:
z = re^{iθ}
= 10 + iy
The complex logarithm volition exist (north = …-2,-i,0,1,two,…):
Log
z =
ln(r) +
i(θ+2nπ)
=
ln(√(x
^{two}+y
^{2})) +
i·arctan(y/x))
Logarithm bug and answers
Problem #one
Find x for
log_{ii}(10) + log_{2}(x-3) = 2
Solution:
Using the product rule:
log_{2}(x∙(x-3)) = 2
Changing the logarithm form co-ordinate to the logarithm definition:
x∙(x-three) = 2^{2}
Or
ten
^{2}-3ten-iv = 0
Solving the quadratic equation:
ten
_{one,2}
= [three±√(ix+16) ] / 2 = [3±5] / two = 4,-1
Since the logarithm is non divers for negative numbers, the answer is:
x
= 4
Trouble #2
Detect x for
log_{iii}(x+two) – log_{3}(10) = 2
Solution:
Using the quotient rule:
log_{three}((ten+2) /
10) = 2
Changing the logarithm grade co-ordinate to the logarithm definition:
(x+2)/ten
= 3^{ii}
Or
x+2 = 9x
Or
8x
= 2
Or
x
= 0.25
Graph of log(x)
log(x) is non defined for real non positive values of x:
Logarithms table
x | log_{
x
} x |
log_{
2
} ten |
log_{
e
} x |
---|---|---|---|
undefined | undefined | undefined | |
^{+} | – ∞ | – ∞ | – ∞ |
0.0001 | -four | -xiii.287712 | -9.210340 |
0.001 | -3 | -9.965784 | -six.907755 |
0.01 | -2 | -half dozen.643856 | -4.605170 |
0.one | -one | -iii.321928 | -2.302585 |
1 | |||
2 | 0.301030 | 1 | 0.693147 |
iii | 0.477121 | 1.584963 | 1.098612 |
four | 0.602060 | 2 | 1.386294 |
5 | 0.698970 | ii.321928 | i.609438 |
six | 0.778151 | 2.584963 | 1.791759 |
7 | 0.845098 | 2.807355 | 1.945910 |
8 | 0.903090 | 3 | 2.079442 |
9 | 0.954243 | three.169925 | 2.197225 |
10 | 1 | 3.321928 | 2.302585 |
20 | ane.301030 | four.321928 | two.995732 |
30 | 1.477121 | four.906891 | 3.401197 |
forty | 1.602060 | 5.321928 | 3.688879 |
fifty | 1.698970 | v.643856 | iii.912023 |
60 | 1.778151 | five.906991 | 4.094345 |
lxx | 1.845098 | 6.129283 | 4.248495 |
lxxx | 1.903090 | 6.321928 | four.382027 |
90 | 1.954243 | 6.491853 | four.499810 |
100 | ii | vi.643856 | 4.605170 |
200 | 2.301030 | seven.643856 | five.298317 |
300 | ii.477121 | 8.228819 | v.703782 |
400 | two.602060 | 8.643856 | v.991465 |
500 | two.698970 | eight.965784 | 6.214608 |
600 | 2.778151 | 9.228819 | 6.396930 |
700 | 2.845098 | 9.451211 | 6.551080 |
800 | 2.903090 | ix.643856 | half-dozen.684612 |
900 | ii.954243 | 9.813781 | 6.802395 |
k | 3 | ix.965784 | vi.907755 |
10000 | 4 | thirteen.287712 | 9.210340 |
Logarithm reckoner ►
See besides
- Logarithm rules
- Logarithm alter of base
- Logarithm of cypher
- Logarithm of i
- Logarithm of infinity
- Logarithm of negative number
- Logarithm calculator
- Logarithm graph
- Logarithm table
- Natural logarithm calculator
- Natural logarithm – ln x
- e constant
- Decibel (dB)
Write how to improve this page
Log 10 X 2 Log 2x 10
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