I've seen a few contalks about logic programming, read a few articles, and aside from the intellectual curiosity in itself I can see that it is very elegant in certain circumstances (although I lack the hands-on experience to feel what those circumstances are).

Since doing a whole program in prolog isn't really practical for me, is there any way to use the logic programming paradigm in mainstream programming languages to deal with specific problems that fit it?

If clojure is your thing there is core-logic https://github.com/clojure/core.logic