There are a variety of ways to find the paths where servers are saving their log files.
1) Ask the owner of the application where they have configured it to store its log files.
2) Check the configuration of the application to see where it stores it log files.
3) Search the internet for the above two items.
4) Learn how to work 'lsof' (See #3, but a simple "sudo lsof | less" and start scrolling down the list can be useful)
5) Poke around for a while - for instance, a lot of interesting things can be found in /var/log on most *nix-based operating systems. In Windows most things generate events that end up in the Event Logs.
Beyond that, a little more information may be helpful. Like, what applications do you have installed?
