That does not define the natural logarithm. It leaves it defined up to an additive constant. You need to define it as a definite integral when the non-convergence of that integral when the upper limit is zero shows that the natural log of zero does not exist.
Take the very basic, high-schoolish, definition of logarithm: , when we already
know-define-demmand that . From this, for some
which, as Drexel already pointed out, is impossible by the very
definition of power of positive real numbers (although he talked about inverse function and
stuff, which isn't necessary in a first approach to this)
Another way to define ln(x) is
You can then show that 0 is not in the domain of ln(x) by showing that does not exist.