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(x)
- Logarithm table
- Logarithm calculator
Logarithm definition
When b is raised to the power of y is equal x:
b^{ y} = x
Then the base b logarithm of x is equal to y:
log_{b}(x) = y
For example when:
2^{4} = 16
Then
log_{2}(16) = 4
Logarithm as inverse function of exponential function
The logarithmic function,
y = log_{b}(x)
is the inverse function of the exponential function,
x = b^{y}
So if we calculate the exponential function of the logarithm of x (x>0),
f (f ^{-1}(x)) = b^{log}b^{(x)} = x
Or if we calculate the logarithm of the exponential function of x,
f ^{-1}(f (x)) = log_{b}(b^{x}) = x
Natural logarithm (ln)
Natural logarithm is a logarithm to the base e:
ln(x) = log_{e}(x)
When e constant is the number:
or
See: Natural logarithm
Inverse logarithm calculation
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 product rule |
log_{b}(x ∙ y) = log_{b}(x) + log_{b}(y) |
Logarithm quotient rule |
log_{b}(x / y) = log_{b}(x) - log_{b}(y) |
Logarithm power rule |
log_{b}(x ^{y}) = y ∙ log_{b}(x) |
Logarithm base switch rule |
log_{b}(c) = 1 / log_{c}(b) |
Logarithm base change rule |
log_{b}(x) = log_{c}(x) / log_{c}(b) |
Derivative of logarithm |
f (x) = log_{b}(x) ⇒ f ' (x) = 1 / ( x ln(b) ) |
Integral of logarithm |
∫ log_{b}(x) dx = x ∙ ( 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}(1) = 0 |
Logarithm of the base |
log_{b}(b) = 1 |
Logarithm of infinity |
lim log_{b}(x) = ∞,when x→∞ |
See: Logarithm rules
Logarithm product rule
The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.
log_{b}(x ∙ y) = log_{b}(x) + log_{b}(y)
For example:
log_{10}(3 ∙ 7) = log_{10}(3) + log_{10}(7)
Logarithm quotient rule
The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.
log_{b}(x / y) = log_{b}(x) - log_{b}(y)
For example:
log_{10}(3 / 7) = log_{10}(3) - log_{10}(7)
Logarithm power rule
The logarithm of x raised to the power of y is y times the logarithm of x.
log_{b}(x ^{y}) = y ∙ log_{b}(x)
For example:
log_{10}(2^{8}) = 8∙ log_{10}(2)
Logarithm base switch rule
The base b logarithm of c is 1 divided by the base c logarithm of b.
log_{b}(c) = 1 / log_{c}(b)
For example:
log_{2}(8) = 1 / log_{8}(2)
Logarithm base change rule
The base b logarithm of x is base c logarithm of x 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}(8) in calculator, we need to change the base to 10:
log_{2}(8) = log_{10}(8) / log_{10}(2)
See: log base change rule
Logarithm of negative number
The base b real logarithm of x when x<=0 is undefined when x is negative or equal to zero:
log_{b}(x) is undefined when x ≤ 0
Logarithm of 0
The base b logarithm of zero is undefined:
log_{b}(0) is undefined
The limit of the base b logarithm of x, when x approaches zero, is minus infinity:
See: log of zero
Logarithm of 1
The base b logarithm of one is zero:
log_{b}(1) = 0
For example, teh base two logarithm of one is zero:
log_{2}(1) = 0
See: log of one
Logarithm of infinity
The limit of the base b logarithm of x, 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 one:
log_{2}(2) = 1
Logarithm derivative
When
f (x) = log_{b}(x)
Then the derivative of f(x):
f ' (x) = 1 / ( x ln(b) )
See: log derivative
Logarithm integral
The integral of logarithm of x:
∫ log_{b}(x) dx = x ∙ ( log_{b}(x) - 1 / ln(b) ) + C
For example:
∫ log_{2}(x) dx = x ∙ ( log_{2}(x) - 1 / ln(2) ) + C
Logarithm approximation
log_{2}(x) ≈ n + (x/2^{n} - 1) ,
Complex logarithm
For complex number z:
z = re^{iθ} = x + iy
The complex logarithm will be (n = ...-2,-1,0,1,2,...):
Log z = ln(r) + i(θ+2nπ) = ln(√(x^{2}+y^{2})) + i·arctan(y/x))
Logarithm problems and answers
Problem #1
Find x for
log_{2}(x) + log_{2}(x-3) = 2
Solution:
Using the product rule:
log_{2}(x∙(x-3)) = 2
Changing the logarithm form according to the logarithm definition:
x∙(x-3) = 2^{2}
Or
x^{2}-3x-4 = 0
Solving the quadratic equation:
x_{1,2} = [3±√(9+16) ] / 2 = [3±5] / 2 = 4,-1
Since the logarithm is not defined for negative numbers, the answer is:
x = 4
Problem #2
Find x for
log_{3}(x+2) - log_{3}(x) = 2
Solution:
Using the quotient rule:
log_{3}((x+2) / x) = 2
Changing the logarithm form according to the logarithm definition:
(x+2)/x = 3^{2}
Or
x+2 = 9x
Or
8x = 2
Or
x = 0.25
Graph of log(x)
log(x) is not defined for real non positive values of x:
Logarithms table
x | log_{10}x | log_{2}x | log_{e}x |
---|---|---|---|
0 | undefined | undefined | undefined |
0^{+} | - ∞ | - ∞ | - ∞ |
0.0001 | -4 | -13.287712 | -9.210340 |
0.001 | -3 | -9.965784 | -6.907755 |
0.01 | -2 | -6.643856 | -4.605170 |
0.1 | -1 | -3.321928 | -2.302585 |
1 | 0 | 0 | 0 |
2 | 0.301030 | 1 | 0.693147 |
3 | 0.477121 | 1.584963 | 1.098612 |
4 | 0.602060 | 2 | 1.386294 |
5 | 0.698970 | 2.321928 | 1.609438 |
6 | 0.778151 | 2.584963 | 1.791759 |
7 | 0.845098 | 2.807355 | 1.945910 |
8 | 0.903090 | 3 | 2.079442 |
9 | 0.954243 | 3.169925 | 2.197225 |
10 | 1 | 3.321928 | 2.302585 |
20 | 1.301030 | 4.321928 | 2.995732 |
30 | 1.477121 | 4.906891 | 3.401197 |
40 | 1.602060 | 5.321928 | 3.688879 |
50 | 1.698970 | 5.643856 | 3.912023 |
60 | 1.778151 | 5.906991 | 4.094345 |
70 | 1.845098 | 6.129283 | 4.248495 |
80 | 1.903090 | 6.321928 | 4.382027 |
90 | 1.954243 | 6.491853 | 4.499810 |
100 | 2 | 6.643856 | 4.605170 |
200 | 2.301030 | 7.643856 | 5.298317 |
300 | 2.477121 | 8.228819 | 5.703782 |
400 | 2.602060 | 8.643856 | 5.991465 |
500 | 2.698970 | 8.965784 | 6.214608 |
600 | 2.778151 | 9.228819 | 6.396930 |
700 | 2.845098 | 9.451211 | 6.551080 |
800 | 2.903090 | 9.643856 | 6.684612 |
900 | 2.954243 | 9.813781 | 6.802395 |
1000 | 3 | 9.965784 | 6.907755 |
10000 | 4 | 13.287712 | 9.210340 |