After a serious google search, I have been unable to find a definite yes/no answer to the following question.
Assume I have c, e and m, and I compute $r = c ^ e \mod m$
This is with regard to the discrete-logarithm problem. My understanding is that revealing r and c does not compromise e, but what about revealing m?
I know enough about cryptography to know that I should never assume anything, thus my reason for asking.
It looks like you are talking about public-key cryptography. For RSA, $e$ and $m$ are the public keys for the recipient; $c$ is the message to be encrypted and presumably is private to the sender, while $r$ is the encrypted message that is presumed to be open to interception. The secret decrypt value is $d:= e^{-1} \bmod$ $\lambda(m)$, which is hard to discover since $m$ is a number that is chosen to be hard to factorize - the product of two large primes.
Even knowing all of $r$ and $c$ for a set of messages (and given that $e$ and $m$ are public anyway), the decrypt key value of $d$ is still secure by the size of the numbers involved.
One of the interesting tricks is to send a message encrypted with your own key but using the decrypt exponent $d$ instead of the public encrpyt $e$. Then anyone can decrypt it but it can only have come from you. Then wrapping this message (suitably tagged) in someone else's public key means only they can read it, and they know that only you could have sent it.