When we are analyzing a document, every segment is 'sent' to the server to see if there are matches. Are any of these lookups written in log files?

If verbose logging is enabled (log into the Administration Console > Tracing > check the "Trace Level" setting), then these lookups are written to the "MultiTrans.Services.Host.log" file on the server. See the following article for information on how to find server logs:
Where are the MultiTrans and Flow server logs located?

If the Trace Level is set INFORMATION, WARNING, ERROR or OFF then these lookups will not be recorded in the logs.