A logo needs to stay sharp at every size, from a favicon to a billboard. That points to one format.
First choice: SVG
SVG is vector — it's drawn from math, so it's perfectly crisp at any scale and the file is tiny. Ideal for logos and icons on the web.
Fallback: transparent PNG
When you need a raster file (for a platform that won't accept SVG), export a transparent PNG at a generous size. It keeps a clean, background-free edge.
Never JPG
JPG has no transparency (you'd get a white box) and adds compression artifacts around sharp edges and text — exactly what a logo can't afford.
Need a raster version? Resize a PNG to the size you need. Related: favicon sizes and transparent formats.