Understanding Logarithms

A logarithm answers the question: what exponent do I need? If b^x = y, then log_b(y) = x. For example, log₂(8) = 3 because 2³ = 8. Logarithms are the inverse operation of exponentiation.

Common Bases

Log base 10 (log or log₁₀) is used in engineering and the Richter scale. Log base e (ln, the natural logarithm) is used in calculus and continuous growth. Log base 2 (log₂) appears in computer science and information theory. All logarithms follow the same rules regardless of base.

Properties

Key properties: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n · log(a), and log_b(b) = 1. The change of base formula lets you convert between bases: log_b(x) = ln(x) / ln(b).

Applications

Logarithms appear in pH calculations (chemistry), decibel scales (sound), Richter scale (earthquakes), population growth modeling, and compound interest calculations. They turn multiplication into addition, which is why they were so valuable before calculators existed.

Use our Logarithm Calculator to compute log values in any common base.