Logarithm Calculator

A logarithm is the exponent you need to raise a base to in order to produce a given number. For example, log_10(1000) = 3 because 10^3 = 1000. This calculator finds that exponent for any base and any positive argument.

ln is the natural logarithm, which uses base e (approximately 2.71828). It appears throughout calculus, physics, and finance - for instance, continuous compound interest uses ln to find doubling time. The calculator treats ln as a special case of the same change-of-base formula it uses for every other base.

No finite exponent on a positive base ever produces zero - the output approaches negative infinity as the input approaches zero, and zero itself has no defined answer. The calculator returns no result for arguments of zero or less.

1 raised to any exponent is always 1, so base-1 logarithms have no defined value for any number other than 1, and even there the answer is ambiguous. The calculator rejects base 1 (and any base of zero or less) as mathematically undefined.

Use the change-of-base formula: log_b(x) = ln(x) / ln(b). The calculator applies this internally for every base, so you can enter any positive base other than 1 directly. If you need to check the result by hand, log_b(x) = log(x) / log(b) works the same way with base-10 logarithms.