Straight lines on the log scale
Exponential regression fits ŷ = a · ebx: the model for anything that grows or decays by a steady percentage. Bacterial colonies, compound interest, radioactive decay, early epidemic spread. The fitting trick is old and clean: take logarithms and the curve becomes a straight line, log ŷ = log a + bx, which ordinary least squares handles directly.
A bacterial culture counted hourly: 52, 84, 130, 205, 330, 520, 830 (thousands). The fit gives a = 52.2 and b = 0.460: about 58.5% growth per hour, a doubling time of ln 2 ÷ 0.460 = 1.51 hours. The log-scale R² is 0.9999, near perfect, as it should be: this colony is doing textbook exponential growth. The model predicts hour 7 at 52.2 · e0.460 × 7 ≈ 1,308 thousand.
Doubling time and half-life
Two readings of b cover most uses. For growth (b > 0), the doubling time ln 2 / b says how long each doubling takes. For decay (b < 0), the same formula with |b| gives the half-life. Both are more quotable than the raw rate: "the culture doubles every hour and a half" lands; "b equals 0.46" does not.
Believing the exponential past its welcome. Real growth hits ceilings: nutrients run out, markets saturate, immunity spreads. An exponential fitted to the early phase predicts the middle badly. When growth starts bending toward a cap, the logistic curve is the natural next model.
Why the R-squared carries a footnote
The fit happens on logged values, so the R² describes the fit there. Back on the original scale, large values dominate the eye and the errors. This is not a flaw; it matches how exponential processes behave, with wobble proportional to size. It does mean the R² here is not directly comparable with one from a straight-line fit.
Frequently asked questions
What is the difference between exponential and linear growth?
Linear growth adds the same amount per step; exponential growth multiplies by the same factor per step. Adding 10 a day is linear. Growing 10 percent a day is exponential, and it outruns any line eventually.
Why must Y be positive?
The fit takes log(Y), and the log of zero or a negative number is undefined. Zero counts usually mean the process has not started or the measurement failed; model those cases separately.
How do I get the growth rate as a percentage?
The per-step percentage change is (e^b - 1) x 100. For b = 0.460 that is 58.5% per step. For small b, b itself is nearly the percentage: b = 0.03 means about 3%.
What if the values decrease?
The same model with b < 0: decay. The calculator reports a halving time instead of a doubling time. Radioactive decay and drug clearance are the classic cases.