Thinking is a process and there are no processes in God. So God doesn’t think, insofar as thought is a process.
That’s a bit of a sweeping statement.
If God doesn’t “think”, then where did the idea of the universe come from? If there are no processes in God, then either the universe always existed, or it just happened without any thought on His part - no plan, no reason, no nothing.
He thinks. How He thinks is anybody’s guess. He’s monitoring the thoughts of 7 billion people simultaneously and making judgements about them, and also their acts, words, and acts they should have done, and didn’t.
He’s sat in judgement on about 150,000 people who died today, died yesterday, and will die tomorrow.
He planned today’s technological world as much as yesterday’s horse and buggy world. He thinks.
He doesn’t change in His essence, but He thinks. He even changes His mind sometimes.
As for logic, it’s a tool He uses, and has given us the ability to use it as well, since we’re made in His image. The universe could be best described as mathematically elegant, or logically elegant.
But I have no more idea of how God uses logic than I have of His ability to simultaneously monitor the thoughts of 7 billion people which must be pretty boring for God sometimes (“What will I have for breakfast?”, “Time to clean my teeth.”, “Damn, I’ve missed the bus!”, “Whoops, my fly’s undone!” and “Jeez, I’m bored.”).
Imagine spending all day monitoring that sort of stuff, non stop, billions of times an hour, for thousands of years. He can have it.