Incentive for block creation in a private blockchain Bitcoin miners are incentivized to produce valid blocks via a trasactional reward. But let's say I wanted to implement blockchain as part of my private data system. Who would my miner's be? I think could just have one automated miner (on a dedicated machine) with the proof-of-work difficulty being set to something very easy. But then I'm wondering, what's the point? Is there anything to gain by using a blockchain like this?

In other words, can someone please give an example---no matter how crude---of how someone could use a private blockchain to log transactions, or whether such a thing has any practical benefit to the users?
The blockchain would no longer be completely decentralized.  You would have a dictator, or a central authority that decides which transactions get processed and which do not.

At this point you could just as well skip the blockchain part of the transaction system as well.