Home › Math & Science › Log Calculator

Log Calculator

Use this log calculator to find logarithms in any base — log base 10, the natural log (ln), log base 2, or a log with a custom base of your choice — and to reverse the process with antilog. Type a number, pick the operation, and the answer appears instantly, with the change-of-base working shown so you can follow the maths.

How it works

A logarithm is the reverse of raising a number to a power. If you know that 10³ = 1000, then log base 10 of 1000 is 3 — the exponent you need. In general, log base b of x is the power y that makes bʸ = x. That single idea covers every kind of log on this page; only the base changes.

Log base 10, written log(x), is the common log used in science and engineering. It counts powers of 10: log₁₀(10) = 1, log₁₀(100) = 2, log₁₀(1000) = 3, and log₁₀(1) = 0. It is the log behind the decibel scale for sound and the Richter scale for earthquakes.

Natural log, written ln(x), uses the base e, about 2.71828. It appears wherever things grow or decay continuously — compound interest, population growth, radioactive decay, and the mathematics of calculus. Ln(10) is about 2.3026, and ln(1) is 0.

Log base 2, written log₂(x), counts powers of 2 and is the natural log for computing: log₂(8) = 3 because 2³ = 8, and log₂(1024) = 10. It measures bits of information and the doubling steps in computer science.

Custom base covers everything else. Pick any positive base other than 1 — 5, 7, 3, or whatever your problem uses — and the calculator applies the change-of-base formula, dividing the base-10 log of x by the base-10 log of the base. The breakdown table shows both of those numbers so you can check the division yourself.

Antilog runs the process backwards. If log₁₀(x) = 3, then x = 10³ = 1000, so the antilog base 10 of 3 is 1000. Antilog base e of a number is simply e raised to that number, and antilog with a custom base raises your chosen base to the number. Unlike a log, an antilog accepts any input, including zero and negative numbers — 10⁻² is 0.01, for example.

Worked example

Find log base 10 of 1000. This is the classic first log problem, and it shows the whole idea in one line.

Step 1: Ask the log question — to what power must I raise 10 to get 1000?
Step 2: Count the powers of 10. 10¹ = 10, 10² = 100, and 10³ = 1000.
Step 3: The power that works is 3, so log₁₀(1000) = 3.
Step 4: Check by going backwards. The antilog base 10 of 3 is 10³ = 1000, which is where you started.

Try it yourself: choose "Log base 10", enter 1000, and press Calculate. The tool gives exactly the same answer — log₁₀(1000) = 3. Then switch to "Antilog base 10", enter 3, and you get 1000 back, confirming that log and antilog undo each other.

FAQ