I made a logo by taking a screenshot and saving it as a .PNG file. The edges on the logo are not as smooth as, say, the Google logo on Google.com.
How can I make the edges smooth?
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In fact, as @Scott said, if you don't have a logo in a vectorial format (or some raw format) it's better to start one from scratch.
As a helping tip, I'd recommend you to use Illustrator to be able to make it in vectorial format so as you can scale it without problems. And, if you let me, I'd use the "Pixel perfect" method, with the option "Snap to pixel grid" from Illustrator.
With this, you won't have problems of scalling and antialiasing and you'll end up with a clear and crispy image (in most cases), because there are not middle pixels.
Hope it helps!