There are two ways to interpret this question. One is to ask whether or not belief in God or the idea of God is logical. The other is to ask if God Himself is a logical being. The answer to both is, “Yes, God is logical,” for similar reasons.
It’s logical to believe in God because the existence of God provides consistent answers to many of the more important philosophical questions asked by human beings. For instance, we know that there cannot be an infinite amount of past time; it is illogical to claim we are currently at the “end” of an infinite string of moments. Therefore, there had to be some beginning moment. Therefore, there has to be something, some cause, that is itself non-caused. Logically, this is God.
Another reason belief in God is logical is that the universe appears to be carefully crafted to allow life. Not only that, but this arrangement allows for the kind of life that is complex enough to be self-aware. Ultimately, there are only two explanations for this: some kind of God or random luck. And punting to “luck” has never been a very logical answer.
The fact that human beings think of “logic” at all also supports the idea of God as a rational belief. If God does not exist, then there is no intelligence, purpose, or meaning in anything. Intelligence, purpose, and meaning are simply illusions created by physics and chemistry. However, if there is no God, it would also mean our reason, intellect, and learning are mere illusions of physics. Lacking God, there is no logic. There is no reason to think our thoughts are meaningful or that they actually reflect reality. Our thoughts may be good for survival or simply the results of randomness, but they can’t be relied on as true. In short, the only way to believe that there is such a thing as logic is for one to believe in some kind of God.
Another key point is that those who believe in the laws of logic are confident in something objective, eternal, and non-material. Any objection that God is immaterial, eternal, or objective would be hypocritical coming from someone who puts stock in the laws of logic.
We also know that God Himself is logical, based on His words and His actions. First and foremost, God acts in a logical way: He plans, communicates, discusses, and acts. God even speaks of “reasoning” with human beings (Isaiah 1:18). He distinguishes between truth and falsehood, a core aspect of basic logic (John 7:18). God does not always act in ways human beings would prefer, but this does not mean His actions are “illogical.”
Also, the fact that there are certain things God “cannot” do, such as lie or change, is evidence of His logical nature. A truly perfect being, by definition, cannot change or he ceases to be perfect. This means God cannot contradict His own nature, or else He would be breaking the laws of logic.
Of course, all of these ideas involving God and logic could be explored in much more detail, but that would require more space than is available here.