There's a family of discrete logarithm problems, one for each representation of a group. (Where I mean "representation" in the usual sense, not the precise mathematical one. It's an important distinction because the secp256k1 group, for instance, is isomorphic to all cyclic groups of the same order, but the discrete logarithm problem on secp256k1 is harder than the additive group on Z/<order of secp256k1>Z, because the isomorphism is computationally intractable.) So there isn't simply one monolithic discrete logarithm problem.