Enter a base and a value to calculate the logarithm.

How to Use the Log Calculator

Base and value are the two fields; leave the base at 10 for a common log or switch it to e for natural log without changing anything else about how the calculator works.

A logarithm asks the reverse question from an exponent: not "what do I get if I raise b to the y," but "what y gets me from b to x." log_b(x) = y means b^y = x, full stop — that's the entire relationship.

Most math libraries, including the one behind this calculator, only compute natural log directly, so any other base runs through the change-of-base formula:

log_b(x) = ln(x) ÷ ln(b)

log₂(8) works out to ln(8) ÷ ln(2) = 3, confirming 2³ = 8. The value has to be positive — no real power of a positive base ever lands on zero or negative, so there's no exponent that would solve the equation for a negative input. Base 10 shows up in engineering and scales like pH; base e (≈2.71828) is calculus's natural log; base 2 belongs to computer science, where it measures binary doublings.