How this calculator works
Take natural logs of every response and fit ln(y) = ln(a) + bx by least squares. Exponentiate the intercept to recover a. The alternative form a × c^x uses c = exp(b). The reported log-scale R squared describes the actual fitting objective; original-scale R squared describes the back-transformed predictions. These predictions are not bias-corrected estimates of a lognormal arithmetic mean.
Formula
ln(y) = ln(a) + bx; a = exp(intercept); c = exp(b); ŷ = a exp(bx) = a c^x.
Worked example
At x = 0, 1, 2 with y = 2, 4, 8, the fit is y = 2 × 2^x. Thus a = 2, b = ln(2), c = 2 and the prediction at x = 3 is 16.
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Frequently asked questions
Can responses be zero or negative?
No. Their real natural logarithms are not defined. Adding an arbitrary offset changes the model.
Is this nonlinear least squares on y?
No. It minimises squared log residuals. Nonlinear least squares on original responses generally gives different coefficients.
What does a negative b mean?
It describes exponential decay, with a growth factor between zero and one per x unit.