So the full backup is reinitialising the log chain and hence the log backups run. It's not a good maintenance schedule though. Switching to simple recovery for index rebuilds is not going to do good things to your ability to restore to a point in time (which I assume is why the DB is in full recovery).
Microsoft Certified Master: SQL Server, MVP, M.Sc (Comp Sci)SQL In The Wild
: Discussions on DB performance with occasional diversions into recoverability
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