Don't let them " take it", it is your property not theirs. Just take it with you every time you fly. After you qualify, don't bother taking it, ( I leave mine at home and fill it out when I get home).
HITC - used to fill out my log book and sign off on everyone of my flights when I was his student. He then gave me my book back for safe keeping.
Some students like to keep their log book at the school, it should be the students choice though.