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Log Calculator

Choose logarithm or antilogarithm, enter a number and base, and get the exact result. Common, natural and base-2 shortcuts are included.

Quick Presets:

Logarithmic & Exponential Calculations

Logarithm Result

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Definition: logₓ(a) answers bˣ = a
Common log: Base 10
Natural log: Base e ≈ 2.71828
Extras: log₂ and antilog included

What is a logarithm?

A logarithm answers the question: to what power must the base be raised to get this number? For example log₁₀(1000) = 3 because 10³ = 1000. It is the inverse operation of exponentiation.

Logarithms compress multiplication into addition: log(x × y) = log(x) + log(y). That property made them the foundation of slide rules and scientific tables before electronic calculators, and it still powers logarithmic scales in sound, earthquakes and pH.

Common, natural and base-2 logs

The common logarithm uses base 10 and is written log or log₁₀. The natural logarithm uses base e (≈ 2.71828) and is written ln. Base-2 logs are essential in computer science for measuring bits and algorithm complexity.

The change-of-base formula, logₓ(a) = log(a) / log(x), lets you compute a logarithm in any base from any other. This calculator applies it automatically for custom bases.

How to use this calculator

  1. Choose Logarithm or Antilogarithm mode.
  2. Enter the number (for a logarithm) or the exponent (for an antilog).
  3. Pick a base: 10, e, 2 or a custom base.
  4. Press Calculate and read the result plus its meaning.

Pro tips

  • Remember log(1) = 0 and log(base) = 1 for any valid base.
  • Use log₂ for computer-science problems like binary search depths.
  • Antilog answers 'the base raised to which exponent gives this value'.

Advantages & considerations

✅ Advantages

  • Any base, including custom values
  • Antilogarithm mode built in
  • Shows the exponential meaning of every result

⚠️ Considerations

  • Input must be positive for logarithms
  • Base cannot be 1 or negative

Frequently asked questions

What is a logarithm?▶
A logarithm is the inverse of a power: logₓ(a) is the exponent you raise the base x to in order to get a. For example log₁₀(1000) = 3.
What is the natural log?▶
The natural log, written ln, uses base e (≈ 2.71828). It appears constantly in calculus, growth models and physics.
How do I calculate log in a custom base?▶
Pick Custom as the base and enter it. The calculator uses the change-of-base formula logₓ(a) = log(a) / log(x).
What is an antilogarithm?▶
The antilogarithm reverses a logarithm: if logₓ(a) = b, then the antilog of b in base x is xᵇ = a. For example the antilog of 3 in base 10 is 1000.
Why must the input be positive?▶
Logarithms of zero or negative numbers are not defined for real numbers, because no power of a positive base produces zero or a negative number.