I keep my paper logbook at home on my desk unless I'm headed to the airport to take a friend for their first flight experience (see "guest book", below), or if I'm headed for a BFR or other instructor flight that needs an endorsement by them in my logbook (haven't forgotten yet, knock on wood).
I use Logbook Pro for my electronic copy (and have it save it's backups to my DropBox, so I have multiple backups on each computer I have DropBox installed on, and in their cloud). Logbook Pro handles making sure my math is right, calculating currency and printing any reports needed for insurance or ratings.
My FBO sends me an electronic copy of my receipt as soon as I check out, so by the time I'm in my car leaving the airport, it's in my inbox. When I get home, I enter it in Logbook Pro and then in my paper logbook. At least once/year I scan my new paper logbook pages and save those to DropBox too.
Why keep both? The electronic copy makes currency and reports super simple, and generates all those reports that otherwise take hours crawling over your logbook. The paper copy is nostalgic, has more details about my flights, has signatures and comments from friends who have flown with me (I use it as a sort of guest book, letting them fill in the flight comments on trips they take with me). It will be passed onto my children or their children when I'm no longer here.