Both are correct but "he's been living under a log" is an idiom to describe someone who doesn't know what's going on or not picking up on things- especially in a social context. I've never heard "beneath" substituted in the case of that idiom. Also you should use "hear it in a song or in a poem." "A" is general, "the" is specific - you would only use "the" if you are referring to a specific poem that we have enough context to assume which one it is.

Beneath is not outdated. If it is rarer — I don’t really think it is — it is only slightly so. “Beneath” is a perfectly modern and commonly used preposition that (usually) means the same thing as “under”.

Both are correct but "he's been living under a log" is an idiom to describe someone who doesn't know what's going on or not picking up on things- especially in a social context. I've never heard "beneath" substituted in the case of that idiom. Also you should use "hear it in a song or in a poem." "A" is general, "the" is specific - you would only use "the" if you are referring to a specific poem that we have enough context to assume which one it is.