The relative path to the unity executable is: "Editor/Unity"
The full path in my case is: "/opt/Unity/Editor/Unity" (just in case you need it)
The default Editor.Log path relative to the user home (e.g. "~/") is: ".config/unity3d/Editor.log"
The full path to the Editor.Log is: "/home/jony/.config/unity3d/Editor.log" (again just in case you need it)
You can take your time with the modifications since the Linux Editor still has quite a few issues and it wont even start when you try to run it on an Ubuntu Server installation (See: http://forum.unity3d.com/threads/qxcbconnection-could-not-connect-to-display.352536/). They might also change other settings like the default location of the Editor.Log etc. since this is still a really early experimental build of the Editor.