I haven't tested this, but this Q seems to have all the config for tracing to a log file using ent lib: Implementing Log file using Enterprise Library in asp.net The relevant section is "listeners/filename"
And to log to another machine, it's just a matter of NTFS permissions and the path
You need to have the code running under a domain account that can write to the remove file (or two local accounts with the same password for a workgroup scenario)
The path will either be \machinename\sharedFolderName or \ipAddress\sharedFolderName or y:\ if you map a letter to the drive.