Yes, you can. In fact you get the copyright of your work the moment you create it in a tangible form. But if you want extra legal benefits in case of a copyright infringement, you should register your copyright and get a strong Proof of Ownership. In CopyrightsWorld, you can register the copyright of your logo as creative expressions of an idea. This does not apply to logos used for business purposes.
As such, a logo can be protected by both copyright and trademark, depending on how it is used.
Copyright protects a creative intellectual design or an expression of an idea, while trademark protects things that identify a business on the marketplace.
This means that you can copyright protect your logo design the moment you create it.
Read more on logo copyright vs trademark