Can of worms alert! Sometimes if you think you won't like the answer, it's best not to ask the question.
I've read the ICO's DP guidance, and done the "self assessment" test. Strictly speaking, if the broadcast log contains information that can be used to identify people (e.g. names!) then it's personal data, and should be registered. I can't see how any of the usual exemptions apply - the charity/not-for-profit one doesn't cover this, nor does the recreational one.
So best not to ask the question.