What Exponential Regression Is
An Exponential Regression Calculator models relationships where the response changes by a constant multiplicative factor per unit increase in x. Two common equivalent forms are:
If B is positive, the curve grows; if B is negative, the curve decays. Exponential models are a strong fit for compounding, population growth, interest-like processes, and many “rate-based” systems.
Why Y Must Be Positive
Standard exponential regression fits a straight line after taking a logarithm of y:
Because log(y) is only defined for y > 0 in real numbers, this tool requires positive y values. If you have zeros or negatives, consider a different model—or a documented shift (y + c) only when it’s justified.
How the Calculator Fits the Curve
The calculator transforms y into log space and runs ordinary least squares:
Then it converts back to the original scale:
With the natural model, base = e. With the base-10 model, base = 10.
Interpreting B: Growth Factor and Percent Change
B is easiest to interpret through the per-unit growth factor:
- Natural model: factor = e^B
- Base-10 model: factor = 10^B
The per-unit percent change is:
Doubling Time and Halving Time
Doubling time answers: “How much does x need to increase for y to double?” For the natural model y = A·e^(Bx):
If B is negative, the model decays—so it’s more meaningful to report halving time:
R², p-values, and Intervals
This calculator reports R, R², and the slope p-value using the log-transformed linear regression (log(y) vs x). Confidence and prediction intervals are computed in log space and then converted back to the y scale, creating multiplicative-style ranges.
FAQ
Exponential Regression FAQs
Answers about positive-y requirements, coefficient meaning, growth rates, doubling time, p-values, and intervals.
Exponential regression fits curves where Y changes multiplicatively with X, such as y = A·e^(Bx) or y = A·10^(Bx). It is commonly used for growth/decay, compounding effects, and processes with constant percentage change per unit of X.
Yes. Standard exponential regression uses log-transformed Y (ln(y) or log10(y)), which requires y > 0. If you have zeros or negative values, you may need a different model or a justified shift (e.g., y+c) with careful interpretation.
A is the value of y when x = 0 (the starting level). B controls the growth/decay rate. Each +1 unit in x multiplies y by e^B. If B > 0 you have growth; if B < 0 you have decay.
In the natural model, the per-unit growth factor is g = e^B. The per-unit percent change is (g − 1)×100%. In the base-10 model, g = 10^B.
Doubling time is the x-change needed to multiply y by 2. For y = A·e^(Bx), doubling time = ln(2)/B (when B > 0). If B < 0, the analogous halving time is ln(2)/|B|.
The slope p-value typically tests H0: B = 0 (no exponential trend). It is computed on the log-transformed linear regression (ln(y) vs x or log10(y) vs x).
Intervals are computed in log space and then converted back to the original scale. This is the standard approach, but it produces multiplicative (log-normal) style intervals and is an approximation on the original y scale.
Yes—if you have summary totals for the transformed response w = ln(y) or w = log10(y): n, Σx, Σw, Σx², Σxw, and Σw². Without log-transformed totals, you need the raw y values.