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Free online logarithm calculator

A logarithm answers the question "what exponent produces this number?" — log_b(x) = y means b^y = x. This calculator computes log in any base and always shows ln, log₁₀, and log₂.

About this calculator

How to use

  1. Enter a positive value (x).
  2. Select a base — 2, e (natural log), 10, or Custom.
  3. For a custom base, enter any positive number other than 1.
  4. The result and expression appear instantly.

The formula

log_b(x) = ln(x) / ln(b)     ← change-of-base formula

Where ln is the natural logarithm. Special cases:

  • log₂(x): used in computer science and information theory
  • ln(x): base e, used in calculus and natural growth models
  • log₁₀(x): common log, used in pH, decibels, and the Richter scale

Source: Abramowitz M, Stegun IA. Handbook of Mathematical Functions. 1964. §4.1

Worked examples

log₂(8) = 3 — because 2³ = 8

log₁₀(1000) = 3 — because 10³ = 1000

ln(e) = 1 — because e¹ = e

log₃(27) = 3 — because 3³ = 27

log₅(25) = 2 — because 5² = 25

Notes

  • log_b(1) = 0 for any base b (because b⁰ = 1).
  • log_b(b) = 1 for any base b (because b¹ = b).
  • Logarithms of values between 0 and 1 are negative.
  • This calculator shows results to 9 significant figures.

Frequently asked

What is a logarithm?
A logarithm is the inverse of exponentiation. log_b(x) = y means b^y = x. For example, log₂(8) = 3 because 2³ = 8.
What is the natural logarithm (ln)?
The natural logarithm uses base e ≈ 2.71828. It appears in calculus, growth models, and compound interest. ln(e) = 1 and ln(1) = 0.
How is the change-of-base formula used?
log_b(x) = ln(x) / ln(b). This lets any logarithm be computed using natural log. For example, log₂(8) = ln(8) / ln(2) = 2.0794 / 0.6931 = 3.
What inputs are invalid?
The value must be positive (x > 0). The base must be positive and not equal to 1. Logarithms of zero or negative numbers are not real numbers.
Can I use a custom base?
Yes. Select "Custom base" from the drop-down and enter any positive number other than 1.

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