A logo is not a brand. People (especially those in business) who look at a logo and call it a brand are misinformed at best.
A logo is part of a brand.
A logo is a visual association, a shorthand if you like to the core of the brand.
I’m going to define brand as a product or a service + experiences, promised, aspirations and emotions connected with it.
A logo identifies, clarifies and defines the product or service it rests on from competitors, partners or similar entities. It creates the quick and handy connection between the viewer and the brand. A visual mnemonic if you like.