I have an old SLR film camera lens with a focal length of 58 mm mounted on my Micro 4/3 camera with a crop factor of 2. How does mounting the lens on a different sized sensor change the effective focal length? I know an x mm lens for my sensor works as 2*x mm fullframe equivalent, but I am not sure about the math in this case.
The focal length of the lens does not change. It remains a 58mm lens. What changes is the field of view the lens yields on the new, smaller format. The field of view will be similar to that a 58x2 => 116mm lens would yield on 35mm film.
Essentially, you're taking the same photo, with the same focal length, but cropping down due to the smaller sensor format.
To get your crop equivalent with a full frame lens, simply multiply the focal length by the crop factor; so for MFT, just double the focal length. Or, in the case of purchasing a native lens, you halve what you would have used on film. That is, if you want the field of view for portraits that a film 90mm f/2 lens yields on film, then for micro four-thirds you're looking for a 45mm lens (e.g., m.Zuiko 45mm f/1.8).
However. Because the focal length is really 45mm, it won't yield as thin a depth of field as the 90mm would have on the larger format.
Focal length is focal length. An "...x mm lens for my sensor..." uses the same way to measure focal length as an x mm lens designed for a FF camera or for any other camera.
When you put a lens of a specific focal length on a camera with a different sized sensor the focal length of the lens doesn't change - the field of view changes because the size of the sensor changed. For any lens when you want to determine the FoV when used on your micro 4/3 camera compared to the FoV of a lens with the same focal length used on a FF camera you need to multiply by the camera's crop factor. The crop factor is the ratio of your camera's linear sensor size compared to the linear measurements of a 36x24mm FF sensor.