Is DISCRETE LOG a NP hard problem?
No one knows, but:


It is suspected that neither factoring nor discrete logarithm are NP-complete, but we have no proof.  (Evidence for the suspicion: they are in NP $\cap$ coNP.  See, e.g., https://cstheory.stackexchange.com/q/159/5038, https://cstheory.stackexchange.com/q/167/5038 for factoring.  It's similarly easy to prove that discrete log is in NP $\cap$ coNP; see, e.g., How hard is finding the discrete logarithm?.  Moreover, if any problem in NP $\cap$ coNP is NP-complete, it follows that NP = coNP.  This would be an unexpected result.)
It is suspected that there is no polynomial-time algorithm for factoring or discrete logarithm, but we have no proof.  (Evidence for the suspicion: we haven't been able to find a polynomial-time algorithm for either, despite a lot of trying.)
There is no known (classical) reduction between discrete log and factoring.  However, I personally wouldn't be shocked if one were found, or if both were found to be instances of some broader problem, or something.  For instance, when we find an algorithm technique that works against one, historically often we've been able to adapt it to the other as well.  So, they seem to be connected or related in some deep way.