Unlike cronie the crontab daemon, the Alpine Linux crond crontab daemon won’t log output of commands in crontab. One way to get around this is to do instead

* * * * * sh -c "cmd 2>&1 | sed -e 's/^/CMDOUT (/' -e 's/$/)/' | logger -t CROND[$(cat /run/crond.pid)]"

so that the output or error message will be displayed as

<time> ... CROND[<pid>] CMDOUT (...)

in the syslog message /var/log/messages.