I just ran into this and am a little disappointed that this has been unanswered for 7 months and isn't already covered by the Alpine wiki.

1. The command is actually rc-update add syslog boot, not syslogd
2. If you haven't already, you need to add the busybox-initscripts with apk add busybox-initscripts (Note: at the time of writing, the package manager for Alpine:3.17 appears to be missing busybox-initscripts, I don't know if this is a temporary issue or not)

I just ran into this and am a little disappointed that this has been unanswered for 7 months and isn't already covered by the Alpine wiki.

1. The command is actually rc-update add syslog boot, not syslogd
2. If you haven't already, on Alpine < 3.17, you need to add openrc and busybox-initscripts with apk add openrc busybox-initscripts. On Alpine >= 3.17, you can add a single package apk add busybox-openrc