Log Formula: Rules, Properties, and Examples

Log Formula

A logarithm tells you the exponent needed to raise a given base to obtain a number. Logarithms are widely used in mathematics, science, finance, and engineering.

Basic Log Formula

The basic logarithmic formula is:

log_b(x) = y

This means:

b^y = x

Here, b is the base, x is the number, and y is the logarithm.

Example of a Logarithm

Consider:

log₂(8) = 3

This is because:

2³ = 8

Product Rule

When multiplying two positive numbers with the same logarithm base:

log_b(xy) = log_b(x) + log_b(y)

For example:

log₂(4 × 8) = log₂(4) + log₂(8)

= 2 + 3 = 5

Quotient Rule

For division, subtract the logarithms:

log_b(x/y) = log_b(x) − log_b(y)

Power Rule

If a number is raised to a power, the exponent can be placed in front of the logarithm:

log_b(xⁿ) = n log_b(x)

Change of Base Formula

A logarithm can be converted to another base using:

log_b(x) = log(x) / log(b)

This formula is useful when a calculator does not have a button for the required logarithm base.

Common and Natural Logarithms

A common logarithm uses base 10 and is written as log(x). A natural logarithm uses the mathematical constant e as its base and is written as ln(x).

log₁₀(100) = 2

ln(e) = 1