I was following until the very end. This felt very handwavy:

>Now, since e is somewhere between 2 and 3, we have to conclude that n must be 2. Further, from our curve, we can see that there can only be one other value of x for which (lnx)/x = (ln2)/2, and since we know that 2⁴ = 4², we know that the other integer must be 4.

Is this sort of thinking something that could be justifiably written in a real proof? Sure, conveniently fishing out 2 as n seems to narrow down the search space for m, but I'd imagine this wouldn't work for harder problems