Redirecting a filehandle is easy especially if you don't care about restoring it later:

open STDOUT, ">>somelogfile" or die "opening logfile: $!\n";
open STDERR, ">&STDOUT" or die "dup'ing STDOUT to STDERR: $!\n";
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Basically, this is the same as example of redirecting STDOUT and STDERR in the docs for open except you don't actually save the filehandles beforehand and restore them afterwards.