centralized simple log server

We have few processes which create logs, say P1, P2 and P3 process create debug-p1.log , debug-p2.log and debug-p3.log respectively.

As now, this processes are crashing the server as it is getting bigger and bigger, so we have 1 big server which send this processes into external server and allow it to complete into that server i.e now P1 can either goes to in server-1 or server-2 or server-3.

As you see, we can’t rely the process execution location but we need its process log. We are thinking to have 1 centralized server which allow us to create and write/append the log. Also, that file is accessible to us from any server.

Any idea how can i ?

PS: As this seems to be most common thing, i prefer not to create just API-service which accepts filename and content.