About the Exponential Regression Calculator
This exponential regression calculator fits a curve to paired x–y data when a straight line will not do. Choose an exponential model y = a·e^(bx) for growth or decay that compounds (populations, viral spread, prices, radioactive decay), a logarithmic model y = a + b·ln(x) for growth that levels off (learning curves, diminishing returns), or a power model y = a·x^b for scaling laws. It reports the fitted coefficients, the equation, R², a prediction at any x and — for exponential fits — the growth rate per step and the doubling time or half-life.
It is designed for students checking TI-84 ExpReg and LnReg results, analysts projecting a trend, and scientists linearizing lab data. The chart overlays your points on the fitted curve, so you can see at a glance whether the model shape suits the data.
Like graphing calculators and spreadsheet trendlines, the fit is done by least squares on transformed data: ln(y) against x for exponential, y against ln(x) for logarithmic and ln(y) against ln(x) for power. Exponential and power models therefore need positive y values, and logarithmic and power models need positive x values; other points are skipped.
How to use the exponential regression calculator
- 1Choose exponential, logarithmic or power, depending on the shape of your data.
- 2Paste the x values and the matching y values in the same order.
- 3Enter an x value to predict.
- 4Read the equation, R² and prediction, and check the chart to confirm the curve fits.
- 5For growth data, use the growth rate and doubling time to describe the trend.
Formula and method
Each model becomes a straight line after a transformation, so ordinary least squares can be used: b = Σ(u − ū)(w − w̄) ÷ Σ(u − ū)² and intercept = w̄ − b·ū, where u and w are the transformed x and y. For the exponential and power models the intercept is ln a, so a = e^intercept. This is the same method used by TI calculators (ExpReg, LnReg, PwrReg) and spreadsheet trendlines, and the reported R² is that of the linearized fit.
For an exponential fit y = a·e^(bx), the equivalent base form is y = a·(e^b)^x, so the growth rate per unit of x is e^b − 1. The doubling time (b > 0) or half-life (b < 0) is ln 2 ÷ |b|. Because the fit minimizes errors in ln(y), it weights small y values relatively more than a nonlinear least-squares fit would.
- a
- Initial value / scale coefficient
- b
- Growth (or decay) constant, or slope on ln x
- e
- Euler’s number, 2.71828…
- R²
- Coefficient of determination of the linearized fit
Worked examples
Population growing about 35% per period
Regressing ln(y) on x gives slope 0.30098 and intercept ln a, so y ≈ 99.92·e^(0.30098x) — equivalently 99.92 × 1.3512^x, a 35.1% rise per period. The doubling time is ln 2 ÷ 0.30098 = 2.30 periods, and the model predicts about 608 at x = 6.
Logarithmic learning curve
Regressing y on ln(x) gives y ≈ 2.001 + 2.017·ln(x) with R² = 0.9997. Gains shrink as x grows: the model predicts 6.65 at x = 10, only about 0.45 above the value at x = 8.
Power-law scaling
Regressing ln(y) on ln(x) gives y ≈ 2.997·x^0.7996. Each doubling of x multiplies y by 2^0.8 ≈ 1.74, and the model predicts about 47.9 at x = 32.
Frequently asked questions
How do you do exponential regression?+
Take the natural log of every y value, run an ordinary linear regression of ln(y) on x, then convert back: the slope is b and e raised to the intercept is a, giving y = a·e^(bx). This calculator does those steps for you and also reports the equivalent form a·(e^b)^x.
When should I use exponential vs. logarithmic regression?+
Use exponential when y grows (or decays) by a roughly constant percentage per step, so the curve gets steeper. Use logarithmic when y rises quickly at first and then levels off, with each doubling of x adding a constant amount. Plot the data and check which curve tracks the points.
Why does exponential regression need positive y values?+
The method fits a line to ln(y), and the logarithm of zero or a negative number is undefined. Exponential curves of the form a·e^(bx) with a > 0 are also always positive, so zero or negative values cannot be on the curve. Shift the data or use a different model if needed.
Is this the same as the TI-84 ExpReg?+
Yes in method: TI-84 ExpReg fits y = a·b^x by regressing ln(y) on x, so its b equals e^b from this calculator’s y = a·e^(bx) form (shown in the equation output). LnReg and PwrReg likewise match the logarithmic and power options here, including the reported r².
How do I find the doubling time from an exponential fit?+
For y = a·e^(bx) with b > 0, the doubling time is ln 2 ÷ b ≈ 0.693 ÷ b in units of x. For decay (b < 0) the same formula with |b| gives the half-life. The calculator reports it automatically for the exponential model.