Logarithm Calculator
Calculate a logarithm to any base, with the change-of-base working shown and log₁₀, ln and log₂ of the same number side by side.
How to use it
- Enter the number, then pick the base — 10 for common logs, e for natural logs, 2 for binary.
- The result is checked by raising the base back to that power.
- Logarithms need a positive number and a base above zero that is not 1.
Examples
| log₁₀(1000) | 3 |
| log₂(1024) | 10 |
A logarithm answers 'what power do I raise the base to, to get this number?' log₁₀(1000) = 3 because 10³ = 1000. This tool takes any base — not just 10, e or 2 — using the change-of-base identity log_b(x) = ln(x) ÷ ln(b), and shows the substitution.
Where each base shows up
| Base | Common name | Used for |
|---|---|---|
| 10 | Common log | pH, decibels, Richter magnitude, orders of magnitude |
| e (≈2.71828) | Natural log | Continuous growth, calculus, compound interest |
| 2 | Binary log | Computer science, information theory, doubling counts |
Worked example
A server load doubles every day; to find how many days until it's 10,240x the current load, solve log₂(10,240) ≈ 13.32 — just over 13 days. The same calculation with a base-10 log gives the number of orders of magnitude instead: log₁₀(10,240) ≈ 4.01, meaning just over four powers of ten.
Questions people ask
What does a logarithm actually tell me?
It answers 'what power do I raise the base to, to get this number?'. log₁₀(1000) = 3 because 10³ = 1000.
Why can't I take the log of a negative number?
No real power of a positive base gives a negative result, so the real logarithm is undefined there. Complex analysis extends it, but not with a single real answer.
Is anything sent to a server?
No. Every calculation runs in your browser, so nothing you type is uploaded and the page keeps working offline once it has loaded.
Also searched for
People find this page looking for calculate log calculator.
Last updated 17 August 2026