No they don’t. Because most of the world don’t use IP addresses that way. And that is one of the several reasons why police here in Finland (apply at least whole Nordic) is not interested about IP — it is useless and doesn’t prove anything and doesn’t pinpoint a person.
And if in your direction ISPs are using IP-addresses differently then I’m not suprised IP4-space is so full.
Of course logging IP like every second is possible. But it is waste of resources for nothing. You should doublechek if demand of that is real, not what you are afrading of.