The logs will just continue. If you have a licensed environment a log will be created per day. When you restart your environment the shutdown and restart of the environment will all be logged in your log file of that day.
The shutdown process will not accept any new database request and it will try to finish te current requests but will shut down after some time even if there are still running processes. So depending on the situation the data will be stored in your database or a role back will be done.