The more "modern" -- like the "dtj" & "K&H" or the "DYD" logos are sometimes called "LogoTypes"... they usually consist of text only.
A logo can sometimes have a shape added, they usually consist of lettering and an recognizable object of some sort.
Then there are the "Cartoon Logos" that take a little longer to produce.
You can see Cartoon logo samples of past projects on this page. Click the logo design samples to view a larger illustrated logo sample. Corporate identity logo design and cartooning is an art. You are buying custom logo art that is going to represent your business, care should be taken.
Quality, professional logo design is not inexpensive, so don't expect to get a quality logo at a discount. You are developing the image for your business, so don't cut back on the quality of the logo just to save a few dollars.