Additional space required for INDEX REORGANIZE MSDN states: 


  ALTER INDEX REORGANIZE; however, log space is required.


The database recovery mode is Simple, does that implies the log space is actually NOT required ?.
To answer more concisely than the existing answers:

REORGANIZE produces the same amount of log with SIMPLE and FULL. It's just that SIMPLE almost immediately makes that space available again.

Except if something is preventing log truncation such as an open transaction (unrelated to the session the REORGANIZE runs in).