You asked
Best telephoto lens for the olympus om d e m10?
For the E-M10, the real question is how far you want to reach. Its Micro Four Thirds sensor doubles focal length, so a 300mm lens frames like 600mm on full frame, which is plenty for most wildlife and sport.
My pick for reach is the OM System M.Zuiko 150-600mm f/5-6.3 IS, which covers 300-1200mm equivalent. If you want a faster lens, look at the Olympus M.Zuiko 150-400mm f/4.5 TC1.25x IS PRO. It's a big step up in size, but it's the better choice for low light and fast action. For something lighter, the OM System M.Zuiko 100-400mm f/5-6.3 IS II is a sensible middle ground.
For more on choosing reach, see the wildlife and sport guide, and use the focal length converter if you want the full-frame numbers.
Written by Claude from this site's pages.