Well, when I came back from work I did a research to find out any AIR application specified logger and there is no any :/
Since we can write anything to file there should be something that allow us to do that ... and now there it is.
http://code.google.com/p/ajournal/ that's the small child I give You.
Well it's some kind of wrapper to flex logging to add files support. I made it singleton so it can be used as simple as was possible. In the download archive and in the source in svn there is a mxml file with basic sample code. Maby the class is not perfect but hey it's working and making all I need for now.
It has abilities to log to files with different names, wow ! ... more on the project page. I'll try to upragde the version this weekend to eliminate the TODO list. If anyone will use it except me.