Well, yes and no. We had (I have 4 of them on my shelf) written log books, but in most squadrons they were maintained by somebody called the Ops Yeoman, i.e., the Admin enlisted person that had no flying experience and was "loaned" to the Ops Department from the Admin Department to do paperwork chores. These folks did their best at filling out log books, but with no clue about PIC, SIC, actual instrument, etc., log books tended to get screwed up. I maintained my own after a while, and spent many hours prior to going into airline interviews cleaning up all the errors from years past.

If it's still done that way in USN squadrons, and you're a Naval Aviator looking to move on to the airline world at some point, you might want to get your log book and maintain it yourself.