Adding to @GDP's answer

    $columns = array('user_id', 'type', 'object', 'lvl', 'date');
    $values  = array();
    foreach ($batch as $row) {
        $array    = array(
            $row->user_id,
            $db->quote($row->type),
            $db->quote($row->object),
            $db->quote($row->lvl),
            $db->quote($row->date),
        );
        $values[] = implode(',', $array);
    }
    $query->insert($db->quoteName('#activity_log'));
    $query->columns($db->quoteName($columns));
    $query->values($values);

    $db->setQuery($query);
    $result = $db->execute();