Sync logs should not be showing in your Inbox, but rather in a sub-folder called Sync Issues. This is not a "mail" folder, and does not show in Outlook when you have the Mail tab selected. It only shows when you are viewing all folders.

http://screencast.com/t/YzI2Y2JiMD

The Sync Logs found in the Sync Issues folder are there for troubleshooting sync errors if you have them, and can otherwise be safely ignored.

For more information on this:
http://office.microsoft.com/en-us/outlook/HP101024401033.aspx