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I am using ZNC but it's not obvious how to make it to log everything and make it searchable.

How on earth could it get more obvious to tick the tickbox labelled 'log'?

How you search the created logfiles is up to you, znc is not a log management system, it merely creates.

Uhm, ZNC runs on the console, there can not be a "tickbox labeled 'log'" because there is no GUI?

znc comes with an web GUI to admin/configure it.