LOGARITHM

Definition of Logarithm

A logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which the base must be raised to produce that number.

Formally, if b^y = x, then the logarithm of x with base b is y, written as:

log_b(x) = y

Example: If 2³ = 8, then log₂(8) = 3.

Properties of Logarithm

• Product Rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient Rule: log_b(x/y) = log_b(x) - log_b(y)
• Power Rule: log_b(x^r) = r × log_b(x)
• Logarithm of 1: log_b(1) = 0
• Change of Base Formula: log_b(x) = log_k(x) / log_k(b)

Natural Logarithm

The natural logarithm is the logarithm to the base e, where e ≈ 2.71828. It is denoted as ln(x). It has important applications in calculus, growth processes, and complex numbers.

Note that ln(1) = 0 because e^0 = 1.