Agree with samuel20 completely, don't leave your mess for someone else to clear up. Any car company buying your car will automatically check if it has finance attached to it (using HPI) so you won't be able to do it that way. Anyone in their right mind buying a car privately should invest that £20 and perform a HPI check for their own protection. If you don't do this then you could end up having to give car back to the finance company who technically will still own the car.

Not too familiar with the legalities of the log book loan, I doubt whether the company providing the loan will have registered it. I don't expect it would show up on an HPI check. You could apply to the DVLA for a new log book, and then sell your car. There are sites such as Car Arena where you receive multiple offers for your car and the dealers will settle your outstanding finance directly and pay the remaining balance to you before they purchase the car. Having never done this with a log book loan I'm not sure about the process and how tied to your car the finance is, other than they have hold of your V5.