The Change of Base Formula

Most calculators only have log₁₀ and ln keys, but logarithms exist for any positive base. The change of base formula bridges the gap: log_b(x) = log_c(x) / log_c(b). This change of base calculator applies the identity with natural logarithms — log_b(x) = ln(x) / ln(b) — which is exact for any base.

Common Bases

Base 10 (common logarithm) is used in scientific notation and decibels. Base e (natural logarithm) is fundamental to calculus and continuous growth models. Base 2 (binary logarithm) dominates computer science — information theory, algorithm analysis, and binary search all use log₂.

Domain Rules

A logarithm log_b(x) is only defined for x > 0, b > 0, and b ≠ 1. The value can be any real number: negative for 0 < x < 1, zero at x = 1, and positive for x > 1. The inverse relationship b^(log_b(x)) = x always holds and is shown as the inverse check in the results.