What is the best way to keep logbook entries backed up?
The FAA requirements are for the information to be logged, without specifying the details of how it is logged. 

It's OK to use electronic log books, and multiple log books. Scanned images are as good as photocopies. Just print them if you ever need them on paper. Some items are required to be signed, so scanned signatures and endorsements may be necessary in an electronic logbook.

The older the information, the less its importance. Log books are normally examined to see if what you're doing is legal, checking ratings, currency, training and reviews, etc. If you are flying without required night or IFR currency, for example, it is conceivable that they'd want to check previous flights to see how often that has happened in the past. 

An insurance company might want to make sure you've got the hours you claim, but the FAA's interest is to see if you are legal.