I have a quantum system that solves a problem that takes $O(MN)$ on a classical computer. However, because it is solved using a quantum algorithm, it takes $O(\log(MN))$.
I also have another algorithm that solves it in $O(N \log(M))$.
So my question is: Since $O(\log(MN)) < O(N \log(M))$, can we say that both algorithms are making "exponential gain"?
I mean, in complexity-context names, what should we call the gain $O(\log(MN))$? and what should we call the gain $O(N \log(M))$?
To be specific, I have an algorithm that measures the distance between M training vector and 1 test vector. Each vector is with N dimensions.