Logarithm Calculator

Calculate logarithms with any base. Get log base 10, natural log (ln), log base 2, and custom base logarithms instantly.

Given a value x and a base b, this calculator returns log base b of x — the exponent b must be raised to in order to equal x — expressed as ln(x) divided by ln(b), the standard change-of-base identity. Alongside the custom-base answer it always computes the natural log (base e), log base 10, and log base 2 of the same value, and those three readouts genuinely do not care what you enter for base: they run straight off x using JavaScript's own Math.log, Math.log10, and Math.log2, never routed through the base field at all. Of the two inputs, base has noticeably more pull on the custom-base result than value does — nudging base by ten percent swings the answer by close to twice the proportional amount that the same ten percent nudge to value does — and the two move the result in opposite directions: raising the base shrinks the exponent needed to reach the same x, so the result falls, while raising the value itself requires a larger exponent to reach it, so the result climbs. There is no dominant single-input relationship for the fixed-base readouts, since they only ever respond to one field, value, and never to base. Left unaddressed here: the calculator has no defined answer for a non-positive value, a non-positive base, or a base of exactly 1, since log base 1 is undefined — every power of 1 equals 1, so no exponent reaches any other number — and in each of those cases the custom-base result silently falls back to zero rather than flagging the input as invalid.