sure... but...
of course you can design your own logo, but that doesn't mean it will be good. usually i cringe when a client tells me they want me to design their logo based on their idea(s). it usually means they have an attachment to it (my daughter drew it in crayon!) that is often not based on anything other than emotion. typically the first idea that i have when working on a branding project never makes it to the first presentation. it takes a lot of research and delving into the brand in order to get a feel for what works. most people who just whip up a logo aren't thinking about the big picture. you can't build a brand with just a logo, either... so that's where having professionals do your work comes into play. just like you can fix your own car, most of the time it's better to have a qualified professional do it.
also, there's the issue of production. i've been handed way too many "final logo files" in microsoft word or JPG that were uneditable/raster.
logo files need to be vector (ie don't use photoshop, use illustrator), then be versioned out to spot color and/or cmyk and rgb/web versions, black, white, 1c versions, etc. and then there's the whole brand guidelines aspect of things. what can / can't a designer do with your logo? just randomly throwing together a logo will not help any designer in the future unless all of these things are considered and done properly.
all that said, yes you can do your own logo, but will you be doing yourself and your business a disservice by doing so?