From what I've seen using Alpine Linux for a few weeks on my Raspberry Pi, logs are written to /var/log/messages after the init process starts and launches the logging service. All logs before the init starts can be retrieved using dmesg? I'm not sure about this though, let me know if I'm wrong.
One of the things I haven't figured out yet is if traditional logging systems can easily do advanced log filtering like showing only logs from the current boot (like -b in systemd), previous boots (-b -1), and showing logs after a specific date and time (--since).