Yep! In Calibre's preferences, go to "Sharing over the net" and click "View server logs". A window appears with two text fields, "Error log" and "Access log". If you want to search/filter the log you'll probably find it easier to copy and paste the text into another text editor first. On my Mac, the log file is also saved as ~/Library/Preferences/calibre/server_access_log.txt.
Note that the log records all sorts of activity besides actual ebook downloads, so you'd want to search for lines containing "/get/epub" or something like that to find the requests you're interested in.
Last edited by anoved; 10-15-2012 at 06:17 PM.
Reason: added where the log file is actually stored