,
or some slight variation on this theme. Although the solution is
certainly correct, the mathematician in me needs to point out that it
requires some substantial mathematics to show that either e or ln(2) is
irrational.
Correct solutions were also received from Abhi Dattasharma, Andreas Low, Yang Lu, Murat Ozkan, Bill Rosenberg, BHActuary, and Sid.
All solutions but one offered ,
or some slight variation on this theme. Although the solution is
certainly correct, the mathematician in me needs to point out that it
requires some substantial mathematics to show that either e or ln(2) is
irrational.
The solution sent in by Abhi Dattasharma was really sweet. Appropriately,
he says saw the answer in 
seconds! Here it is:
is irrational as that e is irrational.
Postscript: After posting the solution above, I received two improvements
based on Dattasharma's solution and requiring no substantial mathematical
knowledge. The first came from Thomas Teo. He observes that if is irrational, then Dattasharma's
solution will suffice. On the other hand if that number is rational,
that makes 
irrational which
when raised to the power of 
will do the trick. Jan Siwanowicz offered a similar improvement.
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