Why so complicated? Why not simply print to STDOUT and directly after that to your file? And after each input, print that input to the file too?

You could write a sub my_print that does the double printing, so instead of:

print "Directory to install to: ";
print $log "Directory to install to: ";
[download]

you would write

my_print("Directory to install to: ");
[download]

and my_print would then be something like:

sub my_print {
  my($txt)= @_;
  print $txt;
  return unless $log;
  print $log $txt;
}
[download]