When using HA, the only way to know which server is processing a specific job is to go into each server's log file and look for log messages about that job; when looking at the logs from the API, there is no way to know.
This information can be useful, so it would be good to add the server name as part of the log message. This can either be done for logs permanently or can be done when a server is aggregating/collating logs from the other servers. The former is probably more efficient than the latter. If we go with the former, I'd say that we should always do it, regardless of HA so the log formatting is consistent and in case that server is added to an HA group later.
For example, instead of this:
we can have this: