I was trying to prove that e is irrational without using the typical series expansion, so starting off   
 $e = a/b $   
 Take the natural log so   
 $1 = \ln(a/b)$   
Then  
 $1 = \ln(a)-\ln(b)$   
So unless I did something horribly wrong showing the irrationality of $e$ is the same as showing that the equation $c = \ln(a)-\ln(b)$ or $1 = \ln(a) - \ln(b)$ (whichever one is easiest) has no solutions amongst the natural numbers. I feel like this would probably be easiest with infinite descent, but I'm in high school so my understanding of infinite descent is pretty hazy. If any of you can provide a proof of that, that would be awesome.

EDIT: What I mean by "typical series expansion" is Fourier's proof <http://en.wikipedia.org/wiki/Proof_that_e_is_irrational#Proof>