If you know the date you want to go back to then you should be able to convert that date into epoch seconds (see Time::Local) and compare with the time last modified (see stat) of your compressed log files to see which one to examine.

If you are using 5.12 or later you should be able to use the built-in IO::Uncompress::Bunzip2 module to read the file. 5.10 or earlier and you will have to use the bunzip2 executable the way JavaFan has shown for gunzip.