Find the logarithm of a number to any base.
log_b(x) is the power you raise b to get x. log₁₀(1000) = 3 because 10³ = 1000.
A logarithm answers the question: to what power must the base be raised to get this number? This calculator finds logarithms to base 10 (common log), base e (natural log, ln), base 2 or any positive base other than 1, shows ln, log₁₀ and log₂ side by side, and has an antilog tool that goes the other way.
Logarithms undo exponents — see the Exponent Calculator — and appear in pH, decibels, earthquake magnitude and compound growth. See also the Scientific Calculator, Half-Life Calculator and Root Calculator.
The calculator applies the standard method step by step.
| Item | Value |
|---|---|
| log₁₀(x) | 3 |
| ln(x) = logₑ(x) | 6.9077552789821 |
| log₂(x) | 9.9657842846621 |
| Your base | 3 |
| Check: base ^ answer | 1000 |
| Logarithm | Value |
|---|---|
| log₁₀(1000) | 3 |
| log₂(8) | 3 |
| ln(e) | 1 |
| log₃(81) | 4 |
| log₁₀(0.01) | -2 |
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 1 | 0.0000 | 0.0000 | 0.0000 |
| 2 | 0.3010 | 0.6931 | 1.0000 |
| 3 | 0.4771 | 1.0986 | 1.5850 |
| 5 | 0.6990 | 1.6094 | 2.3219 |
| 10 | 1.0000 | 2.3026 | 3.3219 |
| 20 | 1.3010 | 2.9957 | 4.3219 |
| 50 | 1.6990 | 3.9120 | 5.6439 |
| 100 | 2.0000 | 4.6052 | 6.6439 |
| 1000 | 3.0000 | 6.9078 | 9.9658 |
The exponent you must raise the base to in order to get a number: log₂(8) = 3 because 2³ = 8.
log usually means base 10 (common log); ln means base e ≈ 2.71828 (natural log).
Product: log(ab) = log a + log b. Quotient: log(a/b) = log a − log b. Power: log(aⁿ) = n log a.
No real power of a positive base gives 0 or a negative number, so the logarithm is undefined.
The inverse of a logarithm: antilog_b(y) = bʸ. The antilog of 3 in base 10 is 1,000.
In pH, decibels, the Richter scale, compound interest, half-life and information theory (bits).
It uses the standard method: log_b(x) = ln(x) ÷ ln(b) if log_b(x) = y then x = bʸ The full working and a worked example are shown on this page.
Decimal results use standard double-precision arithmetic (about 15–16 significant digits). Tools that work with whole numbers — big numbers, factors, bases, combinations — use exact integer arithmetic, so no digits are lost.
Yes for checking your work and learning the method. The steps and formula are shown so you can understand the answer, but follow your teacher's rules about calculators in exams.
No. Everything is calculated in your browser and nothing you enter is uploaded or saved.
