There MIGHT be a conflict if one chose specific definitions from a dictionary, but the general ideas of "logic" and "Religion" are not mutually exclusive. Religion being sets of principles and beliefs, and logic being modes of mental reasoning. The two are even intertwined, as one's worldview informs the sames logic.
According to Einstein, the world needs a start and an end, nothing can be constant which consists of matter. And because Energy=Mass its a never ending circle of nothing until you ave something which does not have to consist of matter e.g, A God. So there, it can be logical, just not on which religion.
Religion cannot be logical because it attempts to provide answers for things that we cannot know. Although we are increasingly finding answers on where we come from, we will never know what will happen when we die. A big reason that people are religious is that it provides people comfort in getting an answer to these questions of afterlife. It is, however, an answer that can never be logical because it has and cannot ever provide evidence to back it up. If religion was logical, it wouldn't be about faith, but knowledge.