By default .NET Agent writes its log to standard output (AKA console) so you can redirect application's standard output to a file.
Also the log has higher chance of helping us if we increase the log level (preferably to Trace). You can configure Agent's log level either via ELASTIC_APM_LOG_LEVEL environment variable which works for any .NET application or if you cannot use environment variables you configure via one of the methods specific to the application framework. At the moment we have alternative configuration sources for applications built using ASP.NET or ASP.NET Core.