Postgresql: Pattern match against array elements?
You can accomplish what you're asking for by doing something like this.

I created a table and data to help better illustrate what I'm doing.

CREATE TABLE logs (id serial NOT NULL PRIMARY KEY, lognum int, flightnum int);

INSERT INTO logs (lognum, flightnum) VALUES (1,6);
INSERT INTO logs (lognum, flightnum) VALUES (1,7);
INSERT INTO logs (lognum, flightnum) VALUES (1,8);
INSERT INTO logs (lognum, flightnum) VALUES (2,80);
INSERT INTO logs (lognum, flightnum) VALUES (3,12);
INSERT INTO logs (lognum, flightnum) VALUES (4,8008);

postgres@[local]:5432:postgres:=# SELECT * FROM logs;
 id | lognum | flightnum 
----+--------+-----------
 13 |      1 |         6
 14 |      1 |         7
 15 |      1 |         8
 16 |      2 |        80
 17 |      3 |        12
 18 |      4 |      8008
(6 rows)

Time: 0.188 ms
postgres@[local]:5432:postgres:=#


Note that 1, 2, and 4 have flights with 8, 80, and 8008.

Now, using this query, find the lognums with 8 as a flightnum. This takes your original query with an array_agg, wraps another query around it to generates subscripts for each member in the array, which can be arbitrarily large. Finally, an outer query wraps that, which uses the generated subscript to allow you to do a comparison against each of the members of the flightnums array to see if they are LIKE '8%'.

SELECT lognum FROM ( 
                      SELECT lognum, 
                      flightnums, 
                      generate_subscripts(flightnums, 1) AS s 
                      FROM ( 
                             SELECT lognum, 
                                    array_agg(flightnum) AS flightnums 
                                    FROM logs 
                                         GROUP BY lognum
                           ) AS t1
                   ) AS t2 
            WHERE flightnums[s]::text LIKE '8%' ORDER BY lognum;


Which gives you the following output

postgres@[local]:5432:postgres:=# SELECT lognum FROM (SELECT lognum, flightnums, generate_subscripts(flightnums, 1) AS s FROM ( SELECT lognum, array_agg(flightnum) AS flightnums FROM logs GROUP BY lognum) AS t1) AS t2 WHERE flightnums[s]::text LIKE '8%' ORDER BY lognum;
 lognum 
--------
      1
      2
      4
(3 rows)

Time: 0.338 ms
postgres@[local]:5432:postgres:=# 


As would be expected from the data above. 

For further array manipulation needs, I would recommend reading up on their chapter in the PostgreSQL documentation arrays