Don't get too excited about the log. It may say absolutely nothing about the ship's company and only state position, speed and course, stocks of coal and water, possibly names of officers joining/leaving the ship. I went looking for one death some time back and - despite it being an officer - not even mentioned by name but the fact they'd sent a burial detail ashore was mentioned. However you will find out where the ship was at the time and what she was doing.
Oher very long shots might include finding the officers' names and investigating whether they had left memoirs or notes or investigating the Imperial War Museum's holdings. There is correspondence there from a very distant relative of mine who was killed in action when HMS
Monmouth was lost with all hands, for example. In this case the Record of Service was very thin indeed: the one-word report "Promising" and then "DD" - Discharged Dead.
I have just discovered that service records are available for RNR Ratings in the series BT377 - see this link
http://www.nationalarchives.gov.uk/records/looking-for-person/royalnavalreserve.htm - so there's another target for you
