Okay, the 2.3 beta version has logging. It's not 100% ready for release, but it's working on my test sites (and live site), so if you want to run with it: http://downloads.wordpress.org/plugin/dreamobjects.zip
There are some other security changes, like after you set your keys, they no longer display in wp-admin (though you can see them in your database). Also there's a 'reset' button which does exactly what it sounds like.
The logging is on the main settings screen and logs to a flat text file. There's really nothing dangerous in it, so leaving it up won't kill you, but it's not something I'd leave forever. When you run a backup, it'll do this:
[2013/01/03 02:33:48] Scheduled ASAP backup.
[2013/01/03 02:33:51] List of files added to the zip.
[2013/01/03 02:33:51] SQL file created (1 MB).
[2013/01/03 02:33:51] SQL filename added to zip.
[2013/01/03 02:33:53] Zip generated (10 MB).
[2013/01/03 02:33:54] Copying backup to DreamObjects.
[2013/01/03 02:33:54] Deleting zip.
[2013/01/03 02:33:54] Deleting SQL dump.
When you save settings it'll do this:
[2013/01/03 02:34:33] Scheduled daily backup. Next backup: January 4, 2013 02:34 pm
Also it no longer kicks off a backup when you save settings (which was killing some people's sites).
Remember, this is the beta release, so I expect to find bugs :/