The nat. log.............e...........is irrational............in a bigger range would smooth out........

where e is Euler's number, a transcendental number approximately 2.718281828. The reason this number e is considered the "natural" base of exponential functions is that this function is its own derivative.^[1][2] Every exponential function is directly proportional to its own derivative, but only when the base is e does the constant of proportionality equal 1.