The obvious thing to check is that the log file path and file name has been configured and that this exists.
It is also worth checking the Log level. If this is set to Warning or Error then it will only write out warnings and errors. Try setting this to Verbose.
The other thing to check is that the user that is running the FastStats Service has access to the Log file location. You can check this from Windows' Services or by looking within the FastStats Configurator (you might need to scroll the window to the right to see the Account field):
The other thing to check is that the connection string to the FS_Config database is not set to use a Trusted Connection but the user that is being used to run the FastStats Service is not permitted to access the FS_Config database within SQL Server. If this is the case then the FastStats Service would not be able to find a valid configuration. In this scenario, you would get an error logged into Windows' Event Viewer.
To check the connection string to the FS_Config database, choose Manage Services and then select Edit database config
If this is set to use a Trusted Connection then try changing this to the 'faststats_service' SQL Server user.
If there are errors reported within Windows' Event Viewer then it is worth ensuring that crash dumps have been configured.
From the File menu of the FastStats Configurator choose User Mode Dump Configuration
Make sure the Dump folder is set to a location to which the faststats service can write and that the Dump Count value is greater than 0.
You will need to restart the FastStats Service in order for the changes to take effect.