What happens when the logbook of an aircraft is lost?
In order to state the plane is airworthy you must be able to show that all AD's have been complied with and that the aircraft is within annual (or 100 hour) inspection; these proofs are typically found in the logbook so no, without the logbook your aircraft is probably not airworthy. It may be possible to fly the plane legally in the US anyway if you receive authorization from your FSDO (which, I believe, would be a "ferry permit").

It's not legally required that your logbook be complete to the beginning of time, just that it shows the current requirements are met (along with the signatures/cert numbers of the AI's that are stating them to be met). However, an incomplete logbook is very important from a resale point of view. Not having complete logs will make a plane very difficult to sell and will certainly affect the price most purchasers would be willing to pay.