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.