FigureCheckExponential and power fits
Exponential fit vs power-law fit: how to choose
Distinguish exponential growth or decay from a power-law relationship using the scientific mechanism, transformed plots, residuals, and valid data domains.
Hands-on example
Try this in FigureCheck
Fit the exponential response, then switch Y to the power response and compare Exponential and Power models on the original scale.
Opens a small synthetic dataset with a step-by-step guide in a new editable session.Exponential and power laws are different models
An exponential model has the form y = aebx + c: equal increases in X multiply the response by a similar factor after accounting for the offset. A power model has the form y = axb + c: scaling X by a factor produces a related scaling of Y.
Use transformations as diagnostics
- An exponential relationship without an offset appears approximately linear when log(Y) is plotted against X.
- A power law without an offset appears approximately linear on a log-log plot.
- An unknown additive offset can break both simple transformation checks.
- Zero and negative values require special care because logarithms are undefined there.
Do not choose a model solely because a transformed graph looks straight. Transformations change the residual scale and implied error assumptions.
Choose using mechanism and residuals
Exponential models often describe constant proportional growth or decay over a suitable interval. Power laws commonly describe scale relationships. Fit both only when both are scientifically plausible, then compare residual structure and validation behavior on the original measurement scale.
Use logarithmic axes correctly and avoid extrapolating far beyond the observed range.
What to report
State the equation, parameter estimates, fitted X range, data transformations, offset treatment, residual behavior, and uncertainty. A high R-squared cannot by itself distinguish exponential from power-law behavior.