I think @SamPhillips's answer covers for what you can do for one process and possibly multiple when you combine logs via the shell redirection he has mentioned.
This prompted me to "log" my thoughts about multi-process in another topic:
One simple way might be a tcp client-server, all processes send their logs to the logging server, it merges them, writes them to one file. (This could make sense if you don't want to use one of the places variants)
In the other topic I also mentioned a few general tools that might be interesting depending on your details.
Lastly there are also a lot of logging services (which may be either interesting or useless depending on what you want and need) and people writing their own logging servers, so that they have something they can run themselves instead of having to use hosted services. Either of those might be useful, if you don't want to write something like a log server yourself.