MySQL

For this challenge we are using MySQL.

          # TABLE SETUP
          CREATE TABLE Logs (
            log_id INT AUTO_INCREMENT PRIMARY KEY
          );
          INSERT INTO Logs (log_id)
          VALUES
            (1),
            (2),
            (3),
            (7),
            (8),
            (10);

          # QUERY
          SELECT start.log_id AS start_id, (
              SELECT min(s.log_id) AS id
              FROM Logs AS s
                LEFT OUTER JOIN Logs AS e ON s.log_id = e.log_id - 1
              WHERE e.log_id IS NULL
                AND s.log_id >= start.log_id
            ) AS end_id
          FROM Logs AS start
            LEFT OUTER JOIN Logs AS end ON end.log_id = start.log_id - 1
          WHERE end.log_id IS null;