What you want to do, I might do this way (with thanks to Illuminatus for the suggested use of "ls -1t"):

chomp( my @Files = `ls -1t $LOGSTR` );
die "No files?" if ! @Files;

# Up to 10G
while ( logsize() > 10_000_000 ) {
    my $doomed = pop @Files;
    warn "DELETE $doomed\n";
    unlink "$LOGSTR/$doomed"
        or warn "Can't unlink '$LOGSTR/$doomed': $!";
    sleep 2;
}

sub logsize {
    my $LOGSIZE = `du -s $LOGSTR`;
    $LOGSIZE =~ s{
                   \A        # start of string
                   ( \d+ )   # any number of digits [$1]
                   \s        # white space
                   .*        # anything
                   \z        # end of string
                 }{$1}xms;

    return $LOGSIZE;
}
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I haven't tested this, however, so use with caution.