You will want to familiarize yourself more with the system() function. It returns the exit status of the given command, not what goes to STDOUT. For that you would use back-ticks or qx//. Also your two chomp lines have normal single-quotes where you need back-ticks.

Generally, I try to solve a problem in terms of the approach proposed by the questioner. But in this case, you have started down a rocky road that will involve the date::manip module or something similar in order to correctly handle months as alpha abbreviations (Jan, Feb, etc.) instead of numbers. It's do-able but more bother than it is worth probably. Here's a more perlish solution:

#!/usr/bin/perl -w
use strict;
my $file;
my $dir_spec = '/home/archive/logs/old';
opendir(LOGDIR, $dir_spec) or die "Can't open $dir_spec: $!\n";
while ( defined($file = readdir(LOGDIR)) ) {
   if (-M "$dir_spec/$file" > 7) {
      unlink("$dir_spec/$file") 
          or die "Can't delete $dir_spec/$file: $!\n";
   }
}
closedir(LOGDIR);
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The key to this is the file test operator -M which returns the number of days old the given file is. Just the thing you needed.