In curve fitting the power exponential curve (you know a better name?) is used for the so-called 'geometric fit'.
The logarithm of the curve is the prime function approximation.
The curve can be generalized to:
A = e and b = 1/2 gives the Stirling curve which is an approximation for the factorial 1). Stirling was the one to discover this relation.
notes
1) Often an expression is given as: