Client logs can be found at this default location:
C:\Program Files (x86)\EA\BioWare\Star Wars - The Old Republic\swtor\retailclient\swtor\logs
Client logs look like this: “client_20110630TO95805.log”.
Thank you for your assistance with this issue.
I attempted to locate the client logs, however I do not have the folder "swtor" under retailclient... the only children folders are LoadingScreens and shaders. I also could not locate them elsewhere in the file system. Though I'm not sure how helpful this will be, here is the log from the launcher.
I also finished attempting the troubleshooting advice, including leaving my computer after hitting Play for 45 minutes, to no avail.