AFAIK, there is no logging for auto archive. We can enable logging in Outlook but reading the log is not possible with publicly available tools. MS has their internal tool to analyze etl(outlook log file format) files.
Btw, what is your intension behind implementing archiving? If your goal is not to save space for exchange servers., then probaby you can give large storage to users and implement message retension in your exchange. This way we can allow users to save more data and at the same time they can recover the accedentally deleted data on their own.