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) ,
Currently, we have around 929 calculators and conversion tables to help you "do the math" quickly in areas such as:
- Free online calculators and tools
- Free online units conversion tools
- Free online web design tools
- Free online electricity & electronics tools
- Mathematics
- Online tool
- Text tool
- PDF tool
- Code
- Ecology
- Numbers
- Algebra
- Trigonometry
- Probability & Statistics
- Calculus & analysis
- Mathematical symbols
- Factorial
- Logarithm
- Logarithm rules
- Logarithm of zero
- Logarithm of one
- Logarithm of Infinity
- Logarithm of negative number
- Logarithm change of base rule
- Derivative of logarithm
- Natural logarithm
- Quadratic equation
And we are still developing more. Our goal is to become the one-stop, go-to site for people who need to make quick calculations or who need to find quick answer for basic conversions.
Additionally, we believe the internet should be a source of free information. Therefore, all of our tools and services are completely free, with no registration required. We coded and developed each calculator individually and put each one through strict, comprehensive testing. However, please inform us if you notice even the slightest error – your input is extremely valuable to us. While most calculators on Justfreetools.com are designed to be universally applicable for worldwide usage, some are for specific countries only.