First of all - are ure sure, that your event system code is hit? (try to write the same text with streamwriter to check).
Secondly - for using Tridion.Logging you need specify logging configuration...examples of configuration you can find in web.cfg or tcmSereviceHost.exe.cfg (section - loggingConfiguration).
For your situation you must have something like this:
<loggingConfiguration name="Logging Application Block" defaultCategory="General" logWarningsWhenNoCategoriesMatch="false">
<listeners>
<add name="Trace File" fileName="C:\Path...\YourFileName.log" formatter="Log Text Formatter" listenerDataType="Microsoft.Practices.EnterpriseLibrary.Logging.Configuration.FlatFileTraceListenerData, Microsoft.Practices.EnterpriseLibrary.Logging, Version=2.0.0.0, Culture=neutral, PublicKeyToken=349a39f202fa9b53" traceOutputOptions="None" type="Microsoft.Practices.EnterpriseLibrary.Logging.TraceListeners.FlatFileTraceListener, Microsoft.Practices.EnterpriseLibrary.Logging, Version=2.0.0.0, Culture=neutral, PublicKeyToken=349a39f202fa9b53" />
</listeners>
<formatters>
<add name="Log Text Formatter" template="{timestamp} <{win32ThreadId}> {message}" type="Microsoft.Practices.EnterpriseLibrary.Logging.Formatters.TextFormatter, Microsoft.Practices.EnterpriseLibrary.Logging, Version=2.0.0.0, Culture=neutral, PublicKeyToken=349a39f202fa9b53" />
</formatters>
<categorySources>
<add switchValue="All" name="Tridion.Logging.LoggingCategory.Tracing">
<listeners>
<add name="Trace File" />
</listeners>
</add>
</categorySources>
<specialSources>
<allEvents switchValue="Information" name="All Events">
<listeners>
<add name="Tridion Event Log" />
</listeners>
</allEvents>
<notProcessed switchValue="All" name="Unprocessed Category" />
<errors switchValue="All" name="Logging Errors & Warnings" />
</specialSources>
BTW - I am not sure, that you need this, and that this a good practice.
Tridion.Logging is used mostly in Tridion Core features, such as services (Batch, TcmServiceHost, Publisher, etc.) and you can specify different logging configuration and different listeners for logging. In your example I think easier just to use some StreamWriter.