I had been running Logos for a long time and had lots of tabs open.
I switched to Firefox and ordered the "12 Days" special of the day.
After completing my purchase, I closed Logos.
I attempted to start Logos back up again and it hung forever at the splash screen. No Logos icon in the tray.
I checked out what it was doing in Process Explorer, but it was completely inactive.
Eventually I had to kill the process. Log files here: 4062.Logos4Log.zip

UPDATE: I looked again at the log files after having killed the process and apparently it was still doing something, as a few more lines had been added. But there was absolutly NO visual feedback to me that anything was still being downloaded and for all practical purposes, even to this techie, it looked like the program had hung. Here are the updated log files: 0724.Logos4Log.zip
I just tried to boot it again and it's hung again. I'm going to leave it overnight and see what happens.