The version open (LOG1, ">>", $logfile) or die "Can't open logfile: $!"; should be correct. If it dies, then the error message is likely to give you useful information. If it doesn't die, then you may want to print to STDERR what $logfile is, because it may be logging to a file you aren't expecting. If that doesn't help, then double-check what directory you are in. If that doesn't work, then triple-check whether something is deleting the file.

At a guess, your problem is that the data is written to a different file or directory than the one you're looking at.