Customarily logos do not have usage restrictions. That somewhat kills the whole idea of a logo.
I've never heard of a designer placing usage restrictions on a logo.
In any event without a contract, no matter what he does, it'll be up to the courts to decide if he's got the right or not.
Typically an artist retains all copyrights to things they create. Whether or not you receive payment has little to do with that. There are only 11 ways you lose your copyright. See here: Is it copyright infringement by US copyright law if someone else modifies and uses my design?Is it copyright infringement by US copyright law if someone else modifies and uses my design? -- In fact this may be a duplicate of your question.
Here are some related questions here:
Should graphic artists be scared at all about anyone stealing their artworks?Should graphic artists be scared at all about anyone stealing their artworks?
What are general practices in handing over usage rights to the client?What are general practices in handing over usage rights to the client?
Charging an Intellectual Property transfer fee for design workCharging an Intellectual Property transfer fee for design work
Copyright ownership: paid by hour vs. paid by projectCopyright ownership: paid by hour vs. paid by project
Charging an Intellectual Property transfer fee for design workCharging an Intellectual Property transfer fee for design work
I, personally, think it's bad practice to charge a client for usage when it comes to a logo. You're essentially holding them hostage.
