But the requirement was first to match sbc/dasds/sdf/dsd/ absolutely and after that anything starts with "modem".
Your code matches the occurance of modem as the last string alone which I believe is not what is required.
my $string ="sbc/dasds/sdf/dsd/modemL" ;
if ($string=~/sbc\/dasds\/sdf\/dsd\/modem.*/) {
print "Matching" ;
}