Enable the logger to log the rc-processes by editing /etc/rc.conf
/etc/rc.conf
rc_logger="YES"
To also log the kernel add klogd.
klogd
# apk add sysklogd # rc-update add klogd boot
You can view the logs in /var/log/dmesg and /var/log/messages.
/var/log/dmesg
/var/log/messages