Here's some code I've used that implements nimdokk's suggestion above:

    open my $log_writer, '>>', $log_file;
    *STDOUT = $log_writer;
    *STDERR = $log_writer;
[download]

where of course you have to set $log_file first. I liked this approach enough to put it in the import sub of a module I named Log::Simple.

I'm sure there are better implementations of this idea on CPAN.