You can sign log messages several ways without blockchain. Even systemd (cough) has support for one crude form, which it calls forward secure sealing. Not sure what proof-of-work, which seems to be the subtext and is really intended for distributed lack of trust, would buy you in this situation.

I can think of a reasonable blockchain-like implementation for logs, actually: chunk up the log, sign, include the hash of the prior sig as the first log message in the new chunk. Boom, blockchain! Throw in Merkle trees somehow and you’ve got yourself a VC-funded startup. You’re welcome.

This would be useful to ensure you aren't missing any log data...an issue near and dear to my heart these days.

Blockchains don’t help with ensuring you’re not missing any data, their purpose is to define which series of logs is the “right one”, which doesn’t make much sense in anything but a distributed context. The purpose of the Bitcoin blockchain is to define the “right” log of transactions, that everyone agrees is the right log.

Indeed. That might actually be a good idea for government records and other legal documents.