The action of you applying for a log book will generate a letter (from the DVLA) to the present owner informing him that someone else has applied for a log book.

Its up to him them to contact the dvla if he contests it in any way.

If the dvla have no contact, they assume he no longer has any legal interest in the vehicle and then issue the log book to whoever applied for it.

All the logbook confirms that you will be the new registered keeper, not neccessarily the legal owner.