Yes! You should register the name of your photography business with the state in which you reside.
Start by visiting the website for your county's Register of Deeds. Do a search to make sure the name you want to use is not already in use. Then, print off the form to register your business name.
After you print it, fill it out completely. They used to require notarization, but as of 9-1-2020, they did not (In NC).
After it is filled out properly, make a copy for yourself, and mail the original to the address listed on the bottom of the form along with the filing fee. As of 9-1-2020, the filing fee is $26 (In NC).
Read the payment information carefully to see what options your county allows you to use for payment.
The last time I registered the name of a business, I forgot to add a check for the filing fee! Thankfully, I remembered the day I mailed it, so I printed another form and sent it in with my payment...don't do what I did!