Quick answer: A logarithm answers the question: to what power must the base be raised to get this number, so log base 10 of 1000 is 3. The natural log uses base e, about 2.718. Key rules: log of a product is the sum of logs, log of a power brings the exponent down, and change of base divides natural logs.
Last reviewed: September 2026
Key facts
- log2(32) = 5 because 25 = 32 — a logarithm asks “to what power?”.
- log10(1000) = 3; ln(20) ≈ 2.996.
- Change of base: logb(x) = ln(x) ÷ ln(b).
- Logs compress huge ranges: pH, decibels and the Richter scale are all logarithmic.
A logarithm is the inverse of an exponent: logb(x) asks “to what power must I raise b to get x?” This free logarithm calculator finds the log of any number in any base — including the natural log (ln, base e) and common log (base 10).
Enter the number and the base. Below you will find the change-of-base formula and where logarithms secretly run the modern world, from pH to earthquakes.
How to use
- Enter the number x (must be positive).
- Enter the base b (positive, not 1).
- The logarithm appears instantly.
- Try base 10, base e (ln) and base 2 to feel the difference.
Logarithm definition and change of base
| Term | Meaning |
|---|---|
| x | The number you take the log of (x > 0). |
| b | The base (b > 0, b ≠ 1). |
| y | The answer: the exponent that makes by = x. |
Logs turn multiplication into addition: log(a×b) = log(a) + log(b) — the trick behind slide rules.
Worked examples
Base 2 and base 10
log2(32): 25 = 32, so the answer is 5 — “32 needs five doublings from 1”.
log10(1000) = 3 because 103 = 1000. Base-10 logs count digits: log10 of a million is 6.
Natural log via change of base
log3(20): ln(20) ÷ ln(3) ≈ 2.9957 ÷ 1.0986 ≈ 2.7268.
Check: 32.7268 ≈ 20. The calculator applies this formula for every base.
Where logarithms hide in real life
| Scale | Formula | Meaning |
|---|---|---|
| pH | −log10[H+] | Each pH unit is 10× acidity |
| Decibels | 10 × log10(power ratio) | +10 dB = 10× the power |
| Richter | log10(amplitude) | +1 = ~32× the energy |
| Bits | log2(possibilities) | n bits store 2n values |
| Compound growth | ln-based | Continuous growth uses e |
Frequently asked questions
What is the difference between ln and log10?
ln is base e (≈ 2.71828), the natural choice for growth and calculus. log10 is base 10, the natural choice for orders of magnitude, pH and decibels.
Why is log of 0 or a negative number undefined?
No power of a positive base gives zero or a negative — by is always positive. The calculator flags these inputs.
What is the change-of-base formula?
logb(x) = ln(x) ÷ ln(b). It lets any calculator that knows ln compute any base — which is exactly what this tool does.
Why do scientists love logarithms?
They compress enormous ranges into readable numbers: pH 1–14 covers a 10-trillion-fold range of acidity.
What is log base 2 used for?
Computing: log2 counts doublings — bits needed, algorithm steps, and “how many times must this double” questions.
Sources
- Khan Academy — Logarithm rules and properties
- Maths is Fun — Logarithms with worked examples
Key takeaways
- logb(x) asks: “b to what power gives x?”
- Any base via change of base: logb(x) = ln(x) ÷ ln(b).
- pH, decibels, Richter and bits are all logarithms in disguise.
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