So say you've got central logging setup at your organization. You're smart and are using rsyslog with SSL/TLS and your own CA. For the most part you can reasonably claim that your log messages are secure from the server that emitted them to the destination in your central logging system but can you guarantee they're not modified after that? No. You can't.
From this perspective using a blockchain for logging security-critical events would be extremely useful. It would be impossible for an attacker to modify the logs after-the-fact so that you could no longer determine which account they used to login.
You wouldn't want to use it for general logging because of the overhead but for things like login events, reboots, etc it would be fantastic.