How to backup full transaction logs for SQL Server 2008 R2
Does the ALTER DATABASE XYZ SET RECOVERY FULL imply that you are normally running in SIMPLE recovery model?  

If so, the logs from the SIMPLE mode period are discarded once they are no longer needed.  There is no time machine to go back and recover the logs that were not saved.  So, to interpret your script:

ALTER DATABASE XYZ SET RECOVERY FULL; -- Begin saving logs
BACKUP DATABASE XYZ TO DISC = foo.bak;-- Full backup gives the log a start point
BACKUP LOG XYZ TO DISC = foo.bak; -- The logs that start with the full can now be backed up.


You can get the full logs by keeping the database in the FULL recovery model at all times.  

Also the statistics are not stored in the log files but in the database.