While I agree that that is probably a good idea, it doesn't answer the question that I asked. It could be that the OP only wants to save one archived log. So, in that case, you could move every file in the glob to a static file name i.e.

foreach my $file (glob('*.log')) {
   rename $file, "oldlog" or die "Oops: $!";
}
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I don't know about you, but I'm not psychic.