Belief in Christianity is not entirely positivistic, but that doesn't prove that it is irrational. Logic weighs many kinds of evidence, not just the physical. Whether or not a belief system is logical depends on how you reach conclusions to questions. Although Christians can not prove the existence of God empirically, they may still have valid beliefs which support their claim.
Perhaps you have not met many logically based Christians, but I would submit that many who rely on empirical evidence are not logically based either. Just because you believe what you read in a text book (or a Bible) does not automatically make you capable of logical thought. Furthermore, positivism (like Christianity) ultimately requires a faith (a belief built on incomplete evidence) that the scientific method is a 100% trustworthy system. Scientific theories also require this kind of faith. The major fallacy I notice in positivists is the same one I see in many Christians - their main response to burden of proof is appeal to authority, not logical reasoning.
There are two ways of arguing this - either you show that Christianity as a contained system has faulty premises or conclusions which don't follow (which you can also do with scientific theories), or you can say that Christianity does not follow from empiricism (which is what I think you're saying).
There are major gaps between Christianity and popular scientific theory. Fortunately, most of these gaps contain no empirical proof to debate (the existence of God, for instance). So now you ask, what about the big bang, evolution, etc? It is helpful that most of the discrepancies between Christianity and positivism exist in the form of scientific theories which can not be 100% proven, and which are still heavily debated within the scientific realm. Again, not much to debate.
Since a simple appeal to authority (either way) will not stand up to scrutiny, we might try to determine which system is most free from faulty premises. The burden of proof rests on the one willing to make that claim. I see no feasible way to make such a measurement, and I would certainly not leap to say positivism as a structure is entirely illogical (although a blind adherence to it would be).
If then, one were to come to the conclusion that both belief systems are equally plausible as self-contained structures, all that is left is to choose sides based on perceived truth. The answer to that question depends on what you're most focused on. Positivism falls greatly short on subjective experience and meaning, which most philosophers, neuroscientists and psychologists will tell you. Christianity falls short on scientific peer review and popular opinion.
Anyway, just trying to show that it IS possible to consider Christianity from a logical perspective.