Fit an exponential model y = a·b^x to your data and assess model quality.
Last updated: March 2026
Use commas to separate x and y. One pair per line. All y values must be positive.
| x | Observed | Predicted | Error | % Error |
|---|---|---|---|---|
| 1 | 2.70 | 2.71 | -0.01 | -0.3% |
| 2 | 7.40 | 7.37 | 0.03 | 0.4% |
| 3 | 20.10 | 20.07 | 0.03 | 0.2% |
| 4 | 54.60 | 54.61 | -0.01 | -0.0% |
| 5 | 148.40 | 148.63 | -0.23 | -0.2% |
Exponential regression is a technique for fitting exponential models to data. Unlike linear regression (which models straight-line relationships), exponential regression captures scenarios where quantities grow or decay at rates proportional to their current size.
The model has the form y = a · b^x, where a is the initial value, b is the growth (if b ≥ 1) or decay (if b ≤ 1) factor, and x is the independent variable. This becomes linear in log-space: ln(y) = ln(a) + x·ln(b), allowing us to use standard linear regression techniques.
Exponential models are ubiquitous in science: population growth, radioactive decay, compound interest, pandemic spread, and Moore's Law all follow exponential patterns. Accurate fitting helps us understand growth rates and make predictions about future values.
We transform the exponential model into a linear one using logarithms:
Bacteria population is measured hourly:
R² = 0.9998 (essentially perfect fit). The exponential model explains 99.98% of the variation in bacterial growth.
If b ≥ 1, you have exponential growth; the quantity increases. If 0 ≤ b ≤ 1, you have exponential decay; the quantity decreases. If b = 1 (or close to it), there's effectively no exponential change.
Growth rate = (b - 1) × 100%. If b = 1.5, the growth rate is 50% per unit. This means the quantity multiplies by 1.5 (grows by 50%) each unit step. For decay, the rate is negative.
R² ranges from 0 to 1. Above 0.9 is excellent; 0.7-0.9 is good; 0.5-0.7 is moderate. Below 0.5 suggests the exponential model may not fit your data well. Try a different model if R² is low.
Exponential functions y = a·b^x produce positive outputs for positive a and b. Negative y values don't exist in the pure exponential model. If you have negative data, use a different model.
Use the equation y = a·b^x with your fitted a and b. Plug in any x value to predict the corresponding y. Note: extrapolation (predicting far beyond your data) is risky and assumes the pattern continues.
If R² is low, try another model: linear (y = mx + b), polynomial (y = ax² + bx + c), logarithmic, or power law. Plot residuals to check for patterns suggesting a better fit.
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