# Assumptions

Suppose we have collected a number of data points which are known to
follow an exponential curve of the form *y* = *a* *e*^{b x}.
That the data is exponential must be derived from the model or by observation.

# Derivation

Given such data, if we take the natural logarithm of both sides of the equation
*y* = *a* *e*^{b x}, we get
log(*y*) = log(*a* *e*^{b x})
= log(*a*) + log(*e*^{b x})
= log(*a*) + *b x*.

If we represent φ = log(y) and α = log(a), then this equation
φ = *α* + *b x*, which is of the appropriate form
for using least squares.

Using least squares, once we find *α*, we may set *a* = *e*^{α}
and therefore we have the coefficients *a* and *b* which describe the exponential
curve which most closely passes through the given data points (*x*_{i}, *y*_{i}) for *i* = 1, 2, ..., *n*.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.