Copyright protects somebody's original work which is in fixed form, but it does not protect colors, typeface, and other similar attributes of the logo, but trademarks do.
If you have designed a unique logo for yourself or any company and you want it not to be used by others without your permission or claiming that this is other's creation, then you need to copyright and also trademark your logo.
Copyright is mostly not used for the logo, but it is for literary or artistic work like books, paintings, etc.