Log Calculator (log, ln, antilog)

Same operation, different bases. log without a written base usually means log_10 (base 10, the common logarithm). ln means log_e (base e, the natural logarithm, where e is about 2.71828). The two are related by a constant: ln(x) = log_10(x) x 2.302585... and log_10(x) = ln(x) / 2.302585. Calculators, decibels, and pH default to log_10; calculus, continuous compounding, and most physics work default to ln because the derivative of ln(x) is the simplest possible expression, 1/x.

log_b(x) = log_k(x) / log_k(b), for any positive k other than 1. This is how you compute a log in a base your calculator does not have a button for. To get log_7(49), pick base 10 (or e) and divide: log_10(49) / log_10(7), which is about 1.6902 / 0.8451, exactly 2. This calculator uses the same identity internally with Math.log (natural log) as the bridge, then echoes the result back in the other common bases so you can verify by inspection.

Match the domain. Base 2 (log_2) for information theory, computer science, and anything counting bits, doublings, or binary partitions (e.g. log_2(1024) = 10 bits). Base 10 (log_10) for decibels, the Richter scale, pH, and order-of-magnitude work in chemistry and engineering. Base e (ln) for anything involving continuous growth or decay, including compound interest taken to the continuous limit, radioactive half-life, and most calculus derivations. Custom bases show up in coding theory and a handful of niche scientific formulas.

Because there is no real exponent that turns a positive base into zero or a negative result. b^x for any positive b > 0 always stays positive and only approaches zero in the limit as x goes to negative infinity. It never actually reaches it, and it never goes negative. Both log_b(0) and log_b(-5) have no real solution, so this calculator rejects them and returns no result instead of NaN. The complex-number version of log does extend the domain, but that is outside the scope of a general-purpose log calculator.

They are inverses of each other. If log_10(1000) = 3, then antilog_10(3) = 10^3 = 1000. The log answers "what exponent gets me from base b to value x"; the antilog answers "what value do I get when I raise base b to exponent x". This calculator's mode toggle flips between the two so you can verify a log by feeding its result back through the antilog and recovering the original input. The dedicated /antilog-calculator/ covers the inverse direction in more depth (large-number scientific notation, presets), while this page focuses on the forward log with change-of-base equivalents.