Exponential log file growth on quiet database with log backups
Have you checked the error log ?
Run select * from sys.databases &amp; look at the log_reuse_wait_desc column.
Run DBCC OPENTRAN and see if you have any open active transactions ?
Query the default trace and get the timings of when the file growths are happening
https://www.mssqltips.com/sqlservertip/3445/using-the-sql-server-default-trace-to-audit-events/