Loona Calculators
Find the logarithm of any number to any base — and get the common log, natural log (ln) and binary log alongside it, each one verified by raising the base back to the answer (the antilog).
A logarithm answers the question: "To what power must I raise the base to get this number?" log10(1000) = 3 because 10 raised to the power 3 is 1000. It is the opposite (inverse) of a power, just as division is the opposite of multiplication.
"log" on most calculators means base 10 — the common logarithm. "ln" means base e (e ≈ 2.71828) — the natural logarithm, used across science and finance. This calculator gives you both, plus base 2 (the binary log used in computing), for any number you enter.
Because no power of a positive base can ever produce 0 or a negative result: 10 to any power is always positive. So for real numbers the logarithm of 0 or a negative number is undefined, and the calculator will tell you so rather than show a made-up answer.
Definition: logb(x) = y exactly when by = x. The logarithm and the power are two sides of the same fact.
Change of base: most calculators only have log (base 10) and ln (base e) keys. To find a log to any other base, divide:
logb(x) = ln(x) ÷ ln(b) = log10(x) ÷ log10(b)
That change-of-base rule is what this page uses internally, so the answer you get for base b is the same one you would get step by step by hand.
Antilog: the antilog of y to base b is simply by — it undoes the logarithm. If log10(1000) = 3, then the antilog of 3 is 103 = 1000. Every result on this page is checked this way before you see it.
Always 0, because any base raised to the power 0 equals 1: b0 = 1. In the same way, logb(b) is always 1, because b1 = b.
Because 1 raised to any power is always 1, so "to what power must I raise 1 to get 100?" has no answer at all — and for the number 1 itself it would have infinitely many. Base 1 is therefore excluded, and so are base 0 and negative bases for real-number work.
Wherever quantities grow by multiplication: earthquake strength (Richter scale), sound loudness (decibels), acidity (pH), compound interest and half-life calculations, and the binary log in computing (how many bits, or how many halvings, a dataset needs).
Add this calculator to your own website or blog — it is completely free, needs no sign-up, and always shows the latest version. Copy the code below and paste it into your page where you want the calculator to appear. Keeping the small “Powered by Loona Automation” credit line under the embed helps keep this tool free — but you are free to style or remove it; the tool works either way.