you should print the captured string instead of $PREMATCH.

# $PREMATCH
# $`

see perlvar (search for "PREMATCH") and perlretut (search for "Extracting matches")

updated solution, the old one was insufficient:

chomp( my $logname = qx{logname} );
my @output = qx{ps -ef | grep $logname};

for my $line ( @output ) {
    if ( $line =~ m{/export/home/$logname/(.+)/bin/} ) {
        print $1, "\n";
    }
}
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update1: edited perldoc hints

update2: updated insufficient code; thanks psini for inspiration of array usage and jethro for pointing out my mistake in the previous code version

update3: fixed typo in code