How to deal with SQL Server Full Text Log Files?
It seems your option is this undocumented stored procedure running in a job:
EXEC sp_fulltext_recycle_crawl_log @ftcat = 'FullTextCatalogName';

Check the Recycle Fulltext Catalog Log Files article by Jonathan Kehayias.
I just tested it in a lab environment and it purges the oldest fulltext log file creating a new live file (in my lab I could see 1 live file and 6 old files, so this must be the default number of files SQL Server keeps).