Inkscape
Inkscape is a free open source SVG vector graphics editing program.
You may scoff at the idea of using this little free program to create your logo, and instead fork over $600 for Photoshop (or raise Adobe's pricing system even higher by pirating it), but understand this. The Scalable Vector Graphics (SVG) file format is the W3C standard for scalable vector graphics (and I follow W3C standards). Many major companies have their logos produced in the SVG format such as Twitter, Microsoft's Windows and Bing, Google's YouTube, Mercedes, BMW, Visa, Mastercard, etc., etc., etc. Many being made in Inkscape as well.
The advantage of the SVG file format is that it can be upscaled to any size with maintaning its mathematically-generated sleek paths and alpha transparency, and these images can be converted into common Web formats or used in large poster-sized print.
Inkscape installation is pretty simple, but also pretty long. It says 170MB for me on Windows. If you would like to save a little space, you won't need to install all those languages that are already checked for you, including Piglatin. If you would like Inkscape to run in Piglatin, by all means, please.
Once it runs you get a plain canvas with quite a lot of sidebar and toolbar icons, but once you play around with the freehand drawing tool, the fill tool and the shapes, the confusion wears off after the first two minutes.
One of the drawing options I like the best is the Bezier curves and straight line tool, but found it hard to control whether it decides to draw straight or curved.
The real fun starts with editing pre-made SVG files. You can easily drag and drop any of the layers, stretch and skew them, and change their fill colour. This is perfect for adding your own branding to, say, social media icons. You can also edit the nodes and radius of all the curves to your heart's content. There's a trace bitmap feature if you are working with a non-SVG image file, but you'll have to tweak the settings to trace more complicated than average graphics.