Yes, but not everyone has access to their webserver's error log file... (think accounts on most hosting providers)

Then create your own logfile. Something like:

BEGIN {
    open STDERR, ">>/home/myowndir/errorlogs.txt";
}
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Or to keep the nice CGI::Carp formatting for logged errors:

BEGIN {
    use CGI::Carp qw(carpout);
    my $fn = "/home/joeschmoe/logs/foo_log";
    open my $fh, ">>", $fn
        or die "Unable to open $fn for appending: $!\n";
    carpout($fh);
}
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Makeshifts last the longest.