How can I log the stdout of a process started by start-stop-daemon?

I am using an init script to run a simple process, which is started with:

start-stop-daemon --start --quiet --chuid $DAEMONUSER    \
    --make-pidfile --pidfile $PIDFILE --background       \
    --exec $DAEMON $DAEMON_ARGS

The process called $DAEMON usually prints log information to its standard output. As far as I can tell this data is not being stored anywhere.

I would like to write or append the stdout of $DAEMON to a file somewhere.

The only solution I know is to tell start-stop-daemon to call a shellscript instead of $DAEMON directly; the script then calls $DAEMON and writes to the logfile. But that requires an extra script which, like modifying the daemon itself, seems the wrong way to solve such a common task.

To expand on ypocat's answer, since it won't let me comment:

start-stop-daemon --start --quiet --chuid $DAEMONUSER    \
 --make-pidfile --pidfile $PIDFILE --background       \
 --startas /bin/bash -- -c "exec $DAEMON $DAEMON_ARGS > /var/log/some.log 2>&1"

Using exec to run the daemon allows stop to correctly stop the child process instead of just the bash parent.

Using --startas instead of --exec ensures that the process will be correctly detected by its pid and won't erroneously start multiple instances of the daemon if start is called multiple times. Otherwise, start-stop-daemon will look for a /bin/bash process and ignore the actual child process running the daemon.