Yeah, this definitely lies outside AEM. I would suggest you use log rotation, with an automated custom script to compact / archive an older file daily. Cron jobs would help if you're on linux, but then you also probably wouldn't use .zip.
Personally, our logs rotate and only the last 5 days are on the server. We do use Loginsight as a log aggregation tool though - it has a java agent that forwards log entries to the central loginsight server automatically, and persists the logs / data for future analysis. I would recommend looking into a log aggregation tool like that one.