Something like this for the last if

# This isn't perlish at all but if you need a faithfull emulation...
my $date = `date '+%m%d%y'|sed s/\ /_/g`;
chomp($date);

print "checking the existence of log";
if (-e "/local/home/admin/log${date}_box_50")
{
    rename "/local/home/admin/temp_log_copy",  "/local/home/admin/log$
+{date}_box_50a"
    or die "Failed to rename";
} else {
    rename "/local/home/admin/temp_log_copy",  "/local/home/admin/log$
+{date}_box_50"
    or die "Failed to rename";
}
[download]

But really this is horrible! Stop trying to program for shell in perl and learn perl properly! I promise you you'll never look back. In fact whenever I'm writing a shell script now-a-days and I'm thinking about a loop or ...| awk {print $1} or something like that I'll reach for perl!