Free online logarithm 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.