You can ask it to provide a prolog or datalog program

Isn't it remarkable that this works? I wouldn't have thought so either, and yet here we are.

Yes, but LLMs are doing other things than just what Prolog does, and that's my point. People don't get excited about Prolog helping them write software, even though it has some capability to do so. LLMs contain other potentially useful algorithms (of not just logic, but epistemology in general, it understands how to create hypotheses and run experiments to confirm or refute them).

Also, of course, any computer program can be considered a logical calculation when interpreted in lambda calculus. And LLMs are computer programs, so you can consider them a formalization of all the above in the formal logic (although not a very good formalization, for various reasons).