The Activity window allows you to;
By default the Activity Window will appear whenever you start a sync. It can also be viewed at any time by choosing; Window > Activity:
To prevent the activity window from appearing automatically;
By default, activity logs are kept for 30 days, after which they are deleted. You can change this behaviour in DropSync > Preferences. Keep in mind that for syncs involving large numbers of files, the activity logs can take up considerable disk space. Activity log files themselves are kept in Library > Application Support > DropSync > Logs