On irrationality of natural logarithm
Is there any rational number $r$ such that ln (r) is rational as well?
If so, what's the proof?
If proofs are too lengthy to be cointained as an answer here, I would truly appreciated any easy-to-understand references to study them.
Solution 1:
Aside from $r=1$, no. To prove it, suppose we had an example. Then we'd write $$\frac mn=e^{\frac ab}\implies e^a=\left( \frac mn \right)^b$$ But, with $a\neq 0$ this would tell us that $e$ was algebraic, which is not the case.