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Logarithm Calculator

log_b(x) for any base — common, natural or custom.

Best for: Working with logarithmic scales, analysing algorithm complexity (O(log n)), modelling growth and decay, and solving exponential equations.

Complete guide

Input

How it's calculated & sources
The Logarithm Calculator uses the standard, deterministic formula, so the same inputs always produce the same result. It runs entirely in your browser, so your inputs are never uploaded. Last reviewed June 2026.

Free & no sign-up · runs entirely in your browser. Results are estimates for general information, not professional advice — verify important decisions with a qualified expert. Last reviewed June 2026.

How it works

log_b(x) = ln(x) / ln(b). Common bases: 10 (log), e (ln), 2 (log₂).

Example

log₁₀(1000) = 3, ln(e) = 1

Frequently asked questions

Restrictions?+

Argument must be positive. Base must be positive and ≠ 1.

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Complete guide

Quick answers

Short, sourced answers to the questions people (and AI assistants) ask most.

What is Logarithm?
A logarithm calculator finds the exponent a base must be raised to in order to produce a given number — written log base b of x.
Why does Logarithm matter?
The log is the power, so it compresses very large ranges into manageable numbers — which is why logs underpin scales like decibels, pH and earthquake magnitude.
How is Logarithm calculated?
log_b(x) = y means b^y = x. The common bases are 10 (written log), e (written ln, the natural log) and 2 (binary). To use any base, apply change of base: log_b(x) = ln(x) / ln(b).
What are common logarithm mistakes?
Taking the log of zero or a negative number — it is undefined for real values.
When should you use the Logarithm Calculator?
Working with logarithmic scales, analysing algorithm complexity (O(log n)), modelling growth and decay, and solving exponential equations.

What is the Logarithm Calculator?

A logarithm calculator finds the exponent a base must be raised to in order to produce a given number — written log base b of x.

How the Logarithm Calculator works

log_b(x) = y means b^y = x. The common bases are 10 (written log), e (written ln, the natural log) and 2 (binary). To use any base, apply change of base: log_b(x) = ln(x) / ln(b).

Logarithms of common values — base 10, natural (ln), base 2

1000
20.3010.69311
e (2.718)0.434311.44
50.6991.612.32
80.90312.083
1012.33.32
161.22.774
10024.616.64
100036.919.97

Worked example

log base 10 of 1000 = 3 (because 10^3 = 1000) log base 2 of 8 = 3 (because 2^3 = 8)

Common mistakes to avoid

  • Taking the log of zero or a negative number — it is undefined for real values.
  • Confusing log (base 10) with ln (base e).
  • Getting the change-of-base ratio upside down.

Keywords: log, logarithm.

Sources: Khan Academy — Logarithms

Reviewed by the Free Tools Galaxy editorial team · Updated June 2026 · Calculated privately in your browser.

Frequently asked questions

Is the Logarithm Calculator free to use?+

Yes. Every tool on Free Tools Galaxy is 100% free, runs in your browser and requires no signup.

How accurate is the Logarithm Calculator?+

Logarithm Calculator uses the standard logarithm formula in double-precision arithmetic, so the same inputs always produce the same result and you can verify any figure by hand. It is an educational estimate — real-world outcomes depend on your actual rates, rules and assumptions.

Do you store my inputs?+

No. The Logarithm Calculator runs entirely in your browser. Nothing is uploaded or saved to a server.

Can I use the Logarithm Calculator on mobile?+

Yes — the interface is fully responsive and works on phones, tablets and desktops.

What are common mistakes to avoid?+

The most frequent mistake is mixing units. Double-check your inputs use a single, consistent unit before clicking Calculate.

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log_b(x) for any base — common, natural or custom.

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The fastest way is to use a free in-browser logarithm calculator: https://freetoolsgalaxy.com/tools/logarithm-calculator. log_b(x) for any base — common, natural or custom.

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