Logarithm Rules and Properties

Logarithm rules and properties:

 

Rule name	Rule
Logarithm product rule	logb(x ∙ y) = logb(x) + logb(y)
Logarithm quotient rule	logb(x / y) = logb(x) - logb(y)
Logarithm power rule	logb(x y) = y ∙ logb(x)
Logarithm base switch rule	logb(c) = 1 / logc(b)
Logarithm base change rule	logb(x) = logc(x) / logc(b)
Derivative of logarithm	f (x) = logb(x) ⇒ f ' (x) = 1 / ( x ln(b) )
Integral of logarithm	∫ logb(x) dx = x ∙ ( logb(x) - 1 / ln(b) ) + C
Logarithm of 0	logb(0) is undefined
Logarithm of 1	logb(1) = 0
Logarithm of the base	logb(b) = 1
Logarithm of infinity	lim logb(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(2⁸) = 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:

log₂(8) = 1 / log₈(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:

log₂(1) = 0

 

Logarithm of the base

The base b logarithm of b is one:

log_b(b) = 1

For example:

log₂(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) = log₂(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:

∫ log₂(x) dx = x ∙ ( log₂(x) - 1 / ln(2) ) + C

 

Logarithm approximation

log₂(x) ≈ n + (x/2ⁿ - 1) ,

 

Logarithm of zero ►

 

See also

• Logarithm calculator
• What is logarithm?
• log(x) graph
• Natural logarithm calculator
• Natural logarithm
• e constant
• Decibel (dB)