The members' names are recorded in the register.
図書館の蔵書登録簿を確認してください。
Please check the library's inventory list.
台帳 is a ledger or account book, typically for financial or property records, whereas 登録簿 is a broader register for any systematic listing.
Compound of 登録 (registration) and 簿 (record book). The kanji 簿 is commonly used in words for official registers and ledgers.