Wensleydale is quite correct. However, museums are often reluctant to take material on long-term loan, not least because they are then providing the owner with free storage. Although nearly all museums have more material than could ever be displayed at one time, those items in store should be viewable on request, and I know that the RAF Museum's archive is made available to researchers, family members etc. Arguably, when a logbook goes into a private collection, it becomes unavailable, unless someone like the OP publishes scans/transcripts on the web.
The decision made to burn unclaimed logbooks in the early 1960s was made by the Air Ministry - ironically, not long before the RAF Museum was founded. The following is from
Air Ministry and Ministry of Defence: Aircrews' Flying Log Books The National Archives - "In 1959 the Particular Instance Paper Committee recommended the destruction, save for a few specimens, of log books remaining unclaimed in the Air Ministry despite announcements in the Press. Further announcements were then made through the Press and BBC stating that books not claimed by 15 September 1960 would be destroyed. Of the 6,400 shelf footage of log books, 20 feet were selected for preservation, aiming to illustrate: format of log bookstypes of flying training at home and abroad, aircrew trades, flying experience, operational flying experience, theatres of war, campaigns and operations. Also selected were: log books of distinguished pilots or members of distinguished squadrons, especially those detailing engagements with the enemy; of pilots with exceptional flying experience; and of foreign nationals flying with the RAF."