This logarithm calculator finds logarithms in any base — log base 10, natural log (ln), log base 2, or a custom base — and can also solve backwards for the number or the base. Enter your values and get the result with the defining equation shown.

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What Is a Logarithm?

A logarithm answers the question “to what power must I raise this base to get that number?” log₁₀(1000) = 3 because 10³ = 1000. Logarithms turn multiplication into addition and powers into multiplication, which is why they powered slide rules, and why they still underpin the pH scale, decibels, Richter magnitudes and computer science. This calculator is part of our Math Calculators collection.

The Three Logarithms You Will Meet

  • Common log — log₁₀(x), written “log”: used in pH, decibels, and scientific scales. log₁₀(1000) = 3 (verified).
  • Natural log — ln(x), base e ≈ 2.718: growth, decay, calculus. ln(e) = 1.
  • Binary log — log₂(x): computing and information. log₂(1024) = 10 (verified) — ten doublings.

Any base works the same way: log_b(x) = ln(x) ÷ ln(b), the change-of-base formula this calculator uses internally. For the inverse direction — raising powers — use the Exponent Calculator; for the huge and tiny numbers logs describe, the Scientific Notation Calculator.

The Log Rules (Product, Quotient, Power)

  • Product: log(a × b) = log a + log b
  • Quotient: log(a ÷ b) = log a − log b
  • Power: log(an) = n × log a
  • log(1) = 0 in every base; log_b(b) = 1.
  • There is no log of zero or a negative number (in real numbers) — no power of a positive base reaches them.

Logarithms in the Wild

  • pH: each pH unit is a 10× change in acidity — our pH Calculator is logarithms at work.
  • Sound: +10 dB means 10× the intensity; +3 dB is roughly double.
  • Algorithms: binary search through a billion items takes about 30 steps because log₂(10⁹) ≈ 30.
  • Earthquakes: each Richter step is ~31.6× the energy of the previous one.

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Frequently Asked Questions

What is log base 10 of 1000?

3, because 10 × 10 × 10 = 1000.

What is the difference between log and ln?

Only the base: “log” usually means base 10, “ln” means base e (2.718…). On some advanced calculators “log” means natural log — always check which convention your course uses.

Why is there no logarithm of 0 or negative numbers?

No real power of a positive base produces 0 or a negative result, so the inverse question has no real answer.

How do I calculate a log with an unusual base?

Change of base: log_b(x) = log(x) ÷ log(b) using any base your calculator has — or pick “custom base” above.

What is the antilog?

The reverse operation: antilog of y in base b is by. Choose “The number x (antilog)” in this calculator.

Is this logarithm calculator free?

Yes — free, instant, no sign-up.