1. What is the relationship?
The Logos is a full-bodied concept. Unfortunately, I will not do it justice nor be able to discuss all its aspects; just a few. Still, they are helpful, especially when considered in light of the others.
I see an aspect of the Logos as an understanding that the Triune God is wise. His mind is perfectly ordered and intellectually beautiful; supreme knowledge without end. I do not believe the concept of rationality even makes sense without this. Rational thought requires using the laws of logic. But the laws of logic are not material, yet they are eternal, unchanging, necessary, immutable, true, and cannot be disproved (for to disprove logic you must employ logic).
But how does logic work in a strictly material universe? If theism is false, from whence does logic arise? It can’t be thought of as merely a descriptive label or just a useful tool or a societal convention or a product of neurochemicals – all of these things negate the concept of logic in some way. If the universe is the product of a gravitational hiccup caused by another universe within the multiverse or if our universe is the result of a chance set of events, first of which is an explosion which defies the law of cause and effect and the principle of sufficient reason, then how does logic hold? It has no real foundation, no place for a source, no ability to be what is must be for the very laws of the universe to work!
Evolution cannot produce real logic because brains just produce what they must produce according to DNA and various chemicals. But this is not what logic is; it is abstract yet real and it is not a material thing – no natural thing produces it, for no natural process on earth even produces new information.
Logic is a description of the way God’s mind works, in a sense, it is how he thinks. Logical thought is a reflection of the completely truthful mind of God Himself. Logic is a mirror on God’s thought process. I am speaking in analogical terms here. But we must see that for scientific investigation to even be possible, the laws of logic must first be in place. Otherwise there is no guarantee so-called natural laws will work, there is no guarantee words make sense, there is no guarantee that the present will be like that past (uniformity of nature), nor is there any guarantee that empirical data is legitimate or sensory perception is actual in any sense.
I know I am making some broad philosophical assumptions and assertions but this argument has been fleshed out in great detail by Christian philosophers such as Greg Bahnsen, Cornelius Van Til, Alvin Plantiga, John Frame, and Gordon Clark (among others). It is sometimes called the Transcendental Argument for God’s Existence. It demonstrates that without God, there is no logic and therefore no science. In essence, the mind of God is what makes science possible.