Exchange 2010 transaction logfiles grow very fast

A few days ago I had a problem with transaction log files on a Exchange 2010 SP2 RU1 mail server. They grown and grown and grown... I saw above 160,000 (one hundred sixty thousand !!!) files belonging to a single mailbox database. The mailbox database itself was ca. 30 GByte in size. It was then no surprise that the appropriate volume exceeded it capacity after only a couple of hours.

I wouldn't even have noticed the issue but the appropriate backup job that should flush transaction log files among other things failed and didn't run. Because of the failed backup job the transaction log files were not flashed and were not removed but grown.