Logarithm

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:

2⁴ = 16

Then

log₂(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^logb^(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

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:

log₁₀(3 ∙ 7) = log₁₀(3) + log₁₀(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:

log₁₀(3 / 7) = log₁₀(3) - log₁₀(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:

log₁₀(2⁸) = 8∙ log₁₀(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:

log₂(8) = 1 / log₈(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 log₂(8) in calculator, we need to change the base to 10:

log₂(8) = log₁₀(8) / log₁₀(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

See: log of negative number

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:

log₂(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:

log₂(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:

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

Logarithm approximation

log₂(x) ≈ n + (x/2ⁿ - 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(√(x²+y²)) + i·arctan(y/x))

Logarithm problems and answers

Problem #1

Find x for

log₂(x) + log₂(x-3) = 2

Solution:

Using the product rule:

log₂(x∙(x-3)) = 2

Changing the logarithm form according to the logarithm definition:

x∙(x-3) = 2²

Or

x²-3x-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

log₃(x+2) - log₃(x) = 2

Solution:

Using the quotient rule:

log₃((x+2) / x) = 2

Changing the logarithm form according to the logarithm definition:

(x+2)/x = 3²

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

Logarithm calculator »