Thats pretty much the default state of any blockchain like systems. You need a private key to write to it. Its just that in most public blockchains can have an infinite amount of new private keys can be generated an and some kind of token is attached to it. For a log none of that would be needed. A central operator could hand out keys to anyone who should be able to write to it and for all other its read-only. And ofc the key alone would still not allow someone to write invalid data.