# Logarithm Rules and Properties

Logarithm product rule

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.

logb(x ∙ y) = logb(x) + logb(y)

For example:

log8(4 ∙ 2) = log8(4) + log8(2)

Logarithm quotient rule

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

logb(x / y) = logb(x) – logb(y)

For example:

log8(5 / 4) = log8(5) – log8(4)

Logarithm power rule

The logarithm of x raised to the power of y is y times the logarithm of x.

logb(x y)y ∙ logb(x)

For example:

log8(35) = 5 ∙ log8(3)

Logarithm base switch rule

The base b logarithm of c is 1 divided by the base c logarithm of b.

logb(c) = 1 / logc(b)

For example:

log3(4) = 1 / log4(3)

Logarithm base change rule

The base b logarithm of x is base c logarithm of x divided by the base c logarithm of b.

logb(x) = logc(x) / logc(b)

For example:

log3(4) = log8(4) / log8(3)

Logarithm of negative number

The base b real logarithm of x when x<=0 is undefined when x is negative or equal to zero:

logb(x) is undefined when ≤ 0

Logarithm of 0

The base b logarithm of zero is undefined:

logb(0) is undefined

The limit of the base b logarithm of x, when x approaches zero, is minus infinity: Logarithm of 1

The base b logarithm of one is zero:

logb(1) = 0

For example:

log4(1) = 0

Logarithm of infinity

The limit of the base b logarithm of x, when x approaches infinity, is equal to infinity:

lim logb(x) = ∞, when x→∞

Logarithm of the base

The base b logarithm of b is one:

logb(b) = 1

For example, the base two logarithm of two is one:

log4(16) = 2

Logarithm derivative

When

f (x) = logb(x)

Then the derivative of f(x):

f ‘ (x) = 1 / ( x ln(b) )

Logarithm integral

The integral of logarithm of x:

∫ logb(x) dx = x ∙ ( logb(x) – 1 / ln(b) ) + C

For example:

∫ log4(x) dx = x ∙ ( log4(x) – 1 / ln(4) ) + C

### Logarithm approximation

log2(x) ≈ n + (x/2n – 1)