create index concurrently if not exists IDX_GIN_LOG_REQUEST on LOG_EXT_SYS using GIN (to_tsvector('english', lower(LOG_REQUEST)));

explain analyze select l__r.LOG_EXT_SYS_ID as __ID
  from LOG_EXT_SYS l__r
 where lower(l__r.LOG_REQUEST) like '%иванов%'

Bitmap Heap Scan on log_ext_sys l__r  (cost=1092.60..7524.54 rows=1626 width=8) (actual time=169.392..1192.972 rows=14010 loops=1)
  Recheck Cond: (lower((log_request)::text) ~~ '%иванов%'::text)
  Rows Removed by Index Recheck: 238
  Heap Blocks: exact=13614
  ->  Bitmap Index Scan on idx_gin_log_request  (cost=0.00..1092.20 rows=1626 width=0) (actual time=160.034..160.034 rows=14248 loops=1)
        Index Cond: (lower((log_request)::text) ~~ '%иванов%'::text)

Planning time: 8.555 ms
Execution time: 1196.950 ms

explain analyze select l__r.LOG_EXT_SYS_ID as __ID
  from LOG_EXT_SYS l__r
 where lower(l__r.LOG_REQUEST) like '%0535537%'

Bitmap Heap Scan on log_ext_sys l__r  (cost=1108.60..7540.54 rows=1626 width=8) (actual time=3248.475..84303.859 rows=3 loops=1)
  Recheck Cond: (lower((log_request)::text) ~~ '%0535537%'::text)
  Rows Removed by Index Recheck: 394733
  Heap Blocks: exact=391763
  ->  Bitmap Index Scan on idx_gin_log_request  (cost=0.00..1108.20 rows=1626 width=0) (actual time=2752.620..2752.620 rows=394736 loops=1)
        Index Cond: (lower((log_request)::text) ~~ '%0535537%'::text)

Planning time: 15.799 ms
Execution time: 84357.465 ms

2)разница в 14248 строк в первом запросе и в 394736 строк во втором - вам ничего не говорит 

create index concurrently if not exists IDX_TRGM_GIN_LOG_REQUEST on LOG_EXT_SYS using GIN (lower(LOG_REQUEST) gin_trgm_ops);