Add to Apache logfile when user leaves webpage

I want to time how long a visitor spends on each page of my website by downloading and analysing the "ex" log file created daily by Apache in the logfiles folder.

I can do this when they move from one page to another but if someone visits the homepage and then leaves I have no idea how long they spent on it.

Could I somehow use Javascript's OnBeforeUnload event to request a file from the server, which would then appear in the log?