How do I find the log for the shutdown process? I've done some Googling and nowhere do I find where you can log the shutdown process.
The best I've found is a log where it states when the shutdown process begins but that's all.

My computer is hanging on the end of the shutdown process and I'm trying to troubleshoot it. 

Have any ideas?
Instead of looking at logs, I would start by booting in verbose mode. (It will also shut down in verbose mode.) Just hold Command ⌘ + V at boot.