Exponential Regression Calculator
Fit y = ae^(bx) to data that grows or decays by a constant percentage — populations, compound growth, radioactive decay, cooling.
Take logarithms of y and the model becomes linear: ln y = ln a + bx.
How the exponential regression calculator works
Take logarithms of y and the model becomes linear: ln y = ln a + bx. Fit a straight line to (x, ln y), then exponentiate the intercept to recover a.
That linearisation is what makes it easy, and it also changes what is being minimised — squared error in log space, which weights small values more heavily than a direct nonlinear fit would. For most purposes it is what you want; the calculator reports R² in both spaces so you can see the difference.
Formula: y = a e^(bx); ln y = ln a + bx
Worked examples
| Inputs | Growth rate b | Note |
|---|---|---|
| 50% growth per step | 0.403453 | b ≈ 0.405, doubling every 1.71 steps |
| Exact doubling | 0.693147 | b = ln 2, doubling time exactly 1 |
| Decay | -0.693147 | b negative, half-life 1 |
FAQFrequently asked questions
When is exponential the right model?
When the *rate of change* is proportional to the current value — populations, compound interest, radioactive decay, epidemics in their early phase.
Why must Y be positive?
Because the fit works on logarithms, and the logarithm of zero or a negative number is undefined.
What is the doubling time?
ln 2 ÷ b. For decay the same formula gives the half-life.
Why are there two R² figures?
Because linearising fits in log space. The log R² is what was minimised; the ordinary R² is how well the curve fits the actual data. They can differ noticeably.
Is exponential growth really unlimited?
Never, in reality. Every real exponential turns over — the logistic curve is usually the honest model once the resource runs short.
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)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.