To list all files that are opened by a *nix process with a given pid, say 42, use the lsof command:
(sudo) lsof -p 42
Of course, a process may have many files open. To list only files that have a name containing "log", use the grep command:
(sudo) lsof -p 42 | grep log
This of course assumes you know the process id (pid) of the process. To find the pid of processes with a given name, e.g. httpd, use the ps command together with the grep command:
# notice that this prints the grep process as well ps aux | grep httpd