I download my dives from my dive computer into MacDive (AWESOME program) as soon as I get home. I like data (not everyone does), so I spend some time with the comments section, writing little journal entries about the dive with as little or as much information as I see fit. For some dives (e.g. unique shipwrecks), I may cut and paste online descriptions of the wrecks into the comment section too. I dive different combinations of backplates, exposure suits and tanks, so recording weight used (if any) can be helpful in the future. Then I sync it with my phone, so I always have all of this information with me at all times.
My log is 100% secure (all of my data is backed up at multiple locations), easily accessible, editable and searchable. If I ever needed a paper copy, I could print out as little or as much as I wanted - but I never have. Not sure about needing to show the log for anything, I have never run into anything like that myself. It's hard to believe that any dive ops in 2014 will still refuse divers because they won't accept digital logs.
These threads are common, and they typically devolve into arguments between people who like to keep logs and people who think that they aren't important. So if you don't care about logging dives, then it doesn't matter. But if you do value your log for whatever reason, I can't for the life of me understand why it would be OK to keep the only copy of this irreplaceable information on a physical paper object that you take with you on a boat.