Last year my fellow students and I spent a goodly portion of our free time considering the similarities of the relationships between sequences and series and those between derivatives and integrals. During the course of our investigations we deduced a sequence form of the exponential function

where

This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithmln , albeit in the sense of being expressed in terms of integers rather than being defined by equations involving sequences and series.

*e*, which stands alone in satisfying the equations^{x}*D f*=

*f*

*f*(0) = 1

where

*D*is the differential operator, producing the derivative of the function to which it is applied.This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithm

*x*

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