I bet you'd have better luck with the first part like so:

open LOG, "home.txt" or die $!;
while(<LOG>) {
    print "$1<br>" if m/\b([\w\d]+)$/;
}
close LOG;
[download]

for the second, I might try something like this

print "$1<br>" if m/\S+\s+[^\/]+\/(\S+)/;
[download]

I'm not sure I understand the problem to be solved in the second part though...