"Logging Out" is an incomplete sentence. You can't say, if it is a past perfect continuous tense, past continuous tense, present perfect continuous tense, present continuous tense, future perfect continuous tense or future continuous tense. Definitely not if you have just these two words. You have to take circumstances into consideration. I mean what question does the "logging out" answer? Is it:
1. What had he been doing?
2. What was he doing?
3. What has he been doing?
4. What is he doing?
5. What will he have been doing? or
6. What will he be doing?

If you resolve this, you'll get an answer what tense, supposedly, the "logging out" is.

My opinion:

It's past continuous tense, question No. 2, "What was he doing?" The "logging out" tells you what was the respective person doing at the time a) you clicked on his profile b) of his last presence on thsi website c) of any other action YOU did that brought you to this kind of information.