You probably need to read the documentation -- although, having just looked at the perldoc for Digest::MD5, it's a little opaque.
You don't need to read anything from the file, or set the binmode .. something like
use Digest::MD5 qw(md5_hex);
open(FILE, "logs.txt"); # YOU NEED ERROR CHECKING HERE
$md5_hex = Digest::MD5->new->addfile(FILE)->hexdigest;
print "MD5: " . $md5_hex . "\n";
will probably work fine. I just deleted unnecessary lines from your code.
Alex / talexb / Toronto
"Groklaw is the open-source mentality applied to legal research" ~ Linus Torvalds