Natural Log Calculator
The natural logarithm — the logarithm to base e, the one that appears wherever growth is continuous.
ln(x) is the logarithm to base e ≈ 2.
How the natural log calculator works
ln(x) is the logarithm to base e ≈ 2.718281828. It is "natural" because it is the one calculus produces on its own: the area under 1/x from 1 to x is exactly ln(x), and e^x is the only function that is its own derivative. That is why continuous growth and decay — compound interest, radioactive half-life, cooling, population — are written with e and inverted with ln.
Formula: log_b(x) = ln(x) / ln(b)
Worked examples
| Inputs | log base b of x | Note |
|---|---|---|
| ln 100 | 4.605170186 | 4.6052 |
| ln e | 1 | 1, by definition |
| ln 2 | 0.6931471806 | 0.6931 — the number behind the rule of 70 |
FAQFrequently asked questions
What is the natural log?
The logarithm to base e ≈ 2.71828, written ln(x).
Why base e?
Because e^x is the only function equal to its own rate of change, so continuous growth is naturally written with it.
What is ln used for?
Continuous compounding, half-lives, cooling curves and anything where the rate of change is proportional to the amount present.
Where does the rule of 70 come from?
ln(2) ≈ 0.693, so doubling time is about 69.3 divided by the growth rate in per cent — rounded to 70 for mental arithmetic.
Is log the same as ln?
No. "log" usually means base 10 and "ln" always means base e — though in pure mathematics "log" often means the natural one.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.