Logarithm Rules and Properties
Logarithm rules and properties:
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 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→∞ |
Logarithm product rule
The logarithm of a 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_{b}(3 ∙ 7) = log_{b}(3) + log_{b}(7)
The product rule can be used for fast multiplication calculation using addition operation.
The product of x multiplied by y is the inverse logarithm of the sum of log_{b}(x) and log_{b}(y):
x ∙ y = log^{-1}(log_{b}(x) + log_{b}(y))
Logarithm quotient rule
The logarithm of a 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_{b}(3 / 7) = log_{b}(3) - log_{b}(7)
The quotient rule can be used for fast division calculation using subtraction operation.
The quotient of x divided by y is the inverse logarithm of the subtraction of log_{b}(x) and log_{b}(y):
x / y = log^{-1}(log_{b}(x) - log_{b}(y))
Logarithm power rule
The logarithm of the exponent 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_{b}(2^{8}) = 8 ∙ log_{b}(2)
The power rule can be used for fast exponent calculation using multiplication operation.
The exponent of x raised to the power of y is equal to the inverse logarithm of the multiplication of y and log_{b}(x):
x ^{y} = log^{-1}(y ∙ log_{b}(x))
Logarithm base switch
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
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)
Logarithm of 0
The base b logarithm of zero is undefined:
log_{b}(0) is undefined
The limit near 0 is minus infinity:
Logarithm of 1
The base b logarithm of one is zero:
log_{b}(1) = 0
For example:
log_{2}(1) = 0
Logarithm of the base
The base b logarithm of b is one:
log_{b}(b) = 1
For example:
log_{2}(2) = 1
Logarithm derivative
When
f (x) = log_{b}(x)
Then the derivative of f(x):
f ' (x) = 1 / ( x ln(b) )
For example:
When
f (x) = log_{2}(x)
Then the derivative of f(x):
f ' (x) = 1 / ( x ln(2) )
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) ,