According to the documentation does:
The server [...] keep a trace of the monitored log's size [...] stored in the Zabbix database [...]

If I'm not wrong storing in database would never be happen for log[*] based items due to lack of mtime (code based on 2.0.8):

1. sed -n '1324,1331p' ./src/libs/zbxdbcache/dbcache.c
   if (-1 == lastlogsize || -1 == mtime)
   continue;

zbx_snprintf_alloc(&sql, &sql_alloc, &sql_offset,
"update items set lastlogsize=" ZBX_FS_UI64 ",mtime=%d where itemid=" ZBX_FS_UI64 ";\n",
lastlogsize,
mtime,
ids[i]);