Log Calculator guide
Log Calculator: Logarithms in Any Base
Use our log calculator to work out logarithms in base 10, base e (natural log), base 2 or any base you choose.
By the AllCalculator editorial teamUpdated 16 October 20261 min read
A logarithm answers the question "what power do I raise the base to, to get this number?" Our Log Calculator works them out in any base.
How to use the calculator (Log Calculator)
- Enter the number.
- Choose the base: 10, e, 2, or another base.
You'll see the result in your chosen base, plus the common log, natural log and binary log.
What a logarithm means
log_b(x) = y means bʸ = x.
So log₁₀(1,000) = 3, because 10³ = 1,000.
| Name | Base | Written |
|---|---|---|
| Common log | 10 | log or log₁₀ |
| Natural log | e ≈ 2.718 | ln |
| Binary log | 2 | log₂ |
Rules of logs
- log(ab) = log a + log b
- log(a ÷ b) = log a − log b
- log(aⁿ) = n log a
- Change of base: log_b(x) = ln(x) ÷ ln(b)
A worked example
With the calculator's default settings, log₁₀(1,000) = 3. The natural log of 1,000 is about 6.908, and log₂(1,000) is about 9.966.
Frequently asked questions
Can I take the log of a negative number?
Not as a real number. Logs are only defined for numbers greater than zero.
Where are logs used?
In the decibel scale for sound, pH in chemistry, the Richter scale for earthquakes, compound interest and computer science.
What's log of 1?
Always 0, in any base, because any number to the power 0 is 1.
To go the other way, use our Exponent Calculator.
Sources and further reading
- GOV.UK: National curriculum in England – mathematicsgov.uk
- Wolfram MathWorld: Logarithmmathworld.wolfram.com
Links open official and independent websites in a new tab.
This guide is general information to help you understand your options. It isn't financial, tax, legal or building advice and doesn't take your personal circumstances into account. Figures were checked when the guide was last updated and can change, so confirm anything important with GOV.UK, HMRC, a regulated adviser or a qualified tradesperson.