I am simply sending the Console log to a text file in the following manner:

DTExec.exe /file "\\%APP_SERVER%\Projects\Runtime\Job1.dtsx" /ConfigFile \\%APP_SERVER%\Projects\Runtime\Job1.dtsConfig.xml /ConsoleLog >> \\%APP_SERVER%\RUNTIME\LOGS\Job1.log

This works fine and and will show any errors, but it also gives a ton of information that I don't need.  Question:  Is it possible to limit the amount of information piped out to the .log file to only the useful information such as an error?

In my opinion enable the logging to file the normal way (via SSIS -> Logging) menu and choose only the events that you deem applicable.

There are options though to control the output via the /Rep[orting] switch, please refer to the dtexec specs https://technet.microsoft.com/en-us/library/ms162810%28v=sql.105%29.aspx

e.g. /Rep E would log errors only