unclebulgaria67 thanks for your reply, but it's sorted now. I just panicked yesterday when I realised that I should have a log book and I didn't have it and I panicked for the reasons you mentioned above.
I called DVLA just now and they said that the car is registered under my name and a log book has been sent to my address, I just didn't receive it for some reason and because I didn't know I was expecting a log book I didn't realise I didn't receive it. So the dealer has done what he had to do, he just didn't tell me they'll send me one so to wait for it. So what happens now is, the DVLA will resend it to me (hopefully I'll get it in 5 working days) and after I receive it I'll fill the new address at Section 6, sign it at Section 8 and send it back to them and they'll send me a new one in 4 weeks time.
Hopefully, it is now sorted! Thanks again.