Logarithm Calculator

A logarithm answers: "What power must I raise the base to, to get this number?" log_b(x) = y means b^y = x. For example, log_2(8) = 3 because 2^3 = 8. log_10(1000) = 3 because 10^3 = 1000. Logarithms are the inverse of exponentiation, just as division is the inverse of multiplication.

log (or log10) uses base 10 — common in science, engineering, and decibels. ln (natural log) uses base e (2.71828...) — fundamental in calculus, physics, and continuous growth. log2 uses base 2 — essential in computer science and information theory. All are related: ln(x) = log(x) / log(e) = log(x) x 2.3026.

Yes — enter any positive base (except 1) and any positive number. The calculator uses the change of base formula: log_b(x) = ln(x) / ln(b). This converts any base to natural logarithm, which JavaScript computes natively. Common bases include 2, e, 10, but any real positive number works.

Key properties: log(a x b) = log(a) + log(b). log(a / b) = log(a) - log(b). log(a^n) = n x log(a). log_b(b) = 1. log_b(1) = 0. These rules let you simplify complex logarithmic expressions and are essential for solving exponential equations.

In real numbers, no power of a positive base gives a negative result. 10^x is always positive for any real x, so log_10(-5) has no real solution. In complex numbers, logarithms of negative numbers exist (involving pi*i), but this calculator works with real numbers only and will indicate when an input is out of domain.