Execute shell scripts with DBMS_SCHEDULER and log output

Question

I have a couple DBMS_SCHEDULER jobs, that executes shell scripts like this:

dbms_scheduler.create_job (
           job_name => 'my_job',
           job_type => 'EXECUTABLE',
           job_action => '/path/to/my_shell.sh',
           enabled => true,
           auto_drop => true,
);

In shell scripts there are some echo's that were logged when calling these shell scripts from crontab. I tried to redirect stdout within Oracle job like this job_action => '/path/to/my_shell.sh >> /path/to/my_log.log', but that didn't work. So, is there a way to achieve this without editing shell scripts?

Thanks in advance!