Not automatically — and this is one of the most commonly misunderstood parts of hiring a designer.
The default assumption is wrong
In most jurisdictions, the creator of an original work holds copyright by default, even after being paid, unless a contract explicitly transfers it. Paying an invoice for design work doesn't, by itself, guarantee you own the resulting artwork — the transfer has to be spelled out.
What a proper agreement should say
Look for language along the lines of "all rights, including copyright, transfer to the client upon receipt of final payment." Without this, you may have a license to use the logo rather than full ownership of it — a meaningful difference if you ever want to modify it, license it, or stop working with that designer.
What "using" a logo without owning it can mean
If only a usage license was granted, you may be restricted in how you can modify the mark, and the original designer could theoretically license or sell a similar design elsewhere. This is rare with professional designers who use standard contracts, but it does happen with informal or undocumented arrangements.
What to do before your next project
Confirm ownership terms in writing before work begins, not after it's finished. If you already have a logo and never signed anything, it's worth reaching out to confirm ownership — most designers will confirm in writing if asked directly.
The short version
Payment alone doesn't guarantee ownership — a written transfer clause does. Confirm it before the project starts, and get it in writing if you haven't already.