No that's not correct, you typically check the alfresco.log file.
You can check the various log file to see if there were any problems in restarting alfresco. But there is no *NEED* for the file to exist prior to starting alfresco they will be created on the fly according to your configuration settings. These are not database transaction log files for example.
To see if alfresco is running you typically just log in and use it.