The easiest way is to insert an image of the logo into a sketch (Tools/Sketch Tools/Sketch Image) and draw on it.
There is also a built-in tool to recognize edges (Autotrace complement as indicated below by @coyote) but it has its limits when you want to have a perfect rendering: a rectangle for example turns into a more or less straight spline depending on the quality of the image.
There is no choice, you have to use illustrator, and then I don't know if it's saving as by choosing the DWG or DXF format, or if you have to export and then you choose the output format, well it's one of the two, but in any case you need adobe illustrator!!