ln (idea)

Definition

The strict definition of the natural logarithm is as an integral: "ln(x) is the integral from 1 to x of dt/t". In functional form:

         / x
        |     dt
ln(x) =  \   ----
          |   t
         / 1

Observations

First, it's clear that ln(1) should be zero. Next, for 0<x<1, the integral is negative (just as it should be). Also, "e^x is the inverse of ln(x)", which is to say that e^ln(x)=x and ln(e^x)=x.
Further, it should now be clear why the harmonic sum (H(x)) diverges; the sum of the reciprocals of the positive integers up to (possibly including) x is an approximation of ln(x) (in the limit as x approaches infinity, H(x)-ln(x)=y, the Euler-Masceroni constant, which is roughly 0.57 or so).

Properties

All the properties that logarithms have are applicable to the natural logarithm, because of the curious property that ln(x)=log_ex (the logarithm to base e of x). Further, the prime number theorem states that:

       pi(x)*ln(x)
lim    -----------  = 1
x->oo       x

There are many other properties of ln (e.g., ln(a*b) = ln(a) + ln(b)), but an exhaustive list would be... exhausting.