This question could be taken in a couple different directions. First, is the idea of God's very existence a logical one? And second, is God Himself a logical being? The answer to both of these questions is "yes." Both the idea of God and the person of God are logical.
Is God logical?
Believing in the existence of God will explain many of the philosophical questions that people have as well as some of the mysteries of the universe. Logically, think of the concept of time—it had to begin at some point. God exists outside of time, so He is the creator of time and space. Look at the precariously delicate balance of earth within the planets and stars, the gravitational pull of the moon, the distance from the sun being just right to sustain human life (Psalm 8:3). Consider human life. The fact that we are self-aware and intelligent seems to be more than simply a lucky coincidence. The design of the universe was no accident. Therefore, it is logical to believe in God as the creator and consistent sustainer of all life.
The act of human beings thinking about the concept of "logic" helps to promote that belief in God is, in fact, rational. Without the existence of God, there is no intentional design, no purpose, no intellect, and no meaning to be found within anything. These things would merely be illusions created by science. Believing in the existence of God gives validity to our intentional design, purpose, intellect, and meaning. We have reason to believe that our thoughts are not only meaningful to us as individuals but also that they truly reflect reality. If we are convinced that there is such a thing as logic, then it is logical to believe that it comes from some type of God.
If we believe in the laws of logic, we are convinced that something objective, non-material, and eternal exists. Objecting to an eternal God who is immaterial and objective would essentially be hypocritical coming from one who believes in the laws of logic.
The actions of God, evident throughout Scripture, show that He Himself is a logical being. He communicates, makes plans, has discussions, and acts accordingly. He is Truth and differentiates between what's true and false, an element of basic logic (John 7:18). He invites humans to reason with him: "Come now, let us reason together, says the LORD" (Isaiah 1:18).
God Himself created logic. Sometimes God's actions do not make sense to humans, but this does not make them illogical. We can find concrete evidence of God's logical nature in that He limits Himself. There are some things, such as lying or changing, that He "cannot" and will not do (Titus 1:2; Malachi 3:6). God and His ways are perfect (Psalm 18:30), so in order to maintain perfection, He cannot change. If God contradicted His own nature, that would be illogical.
There is much more that we could go into regarding the logical aspects of God, but for the sake of length, this will have to do for now. May you continue your study to better understand the logic of believing in God and His logical nature.
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God is spirit – What does that mean?
Is God a person?
Can the existence of God be proven?
Truth about God