Fog can be deceptive. As it happens with other weather phenomenons, the way our eyes and mind perceive a foggy landscape is different from what the camera will capture.
One of the reasons is that contrast can be tricky to find in fog. As we know, contrast is really important in black and white photography, so we need to know how to create it in foggy conditions.
Far away subjects might look attractive to our eyes. I captured these trees above with my telephoto lens, it was a beautiful scene and one would think it'd create a beautiful image. Nothing further from the truth: there's no contrast in the image, only a faint subject.
In foggy conditions, the further the object, the lighter it will appear; the closer it is, the darker it will be. This makes fog the perfect medium to create layers in our images, mixing dark and light objects comes down to close and far objects.
This means you will have to get close to your subject and use a relatively wide lens. The closer you get, the darker and clearer it will be. Then we can use the fog to contrast it against either emptiness, or a faded background. A telephoto won't work: it can't get through all that fog to create enough contrast.
Don't throw away that expensive telephoto lens, though: you can use it to hunt for objects that rise above or through the fog. Ideally, you'll be at a higher position and will look for peaks, trees, power lines and whatever that can make it through the thick fog.
I love fog, it makes everything so beautiful. But in doing so, it can trick us into thinking we can shoot whatever we see. Don't fall into this temptation and look for the contrast that will make for a good image. See through the fog.
Might interest you: 5 ideas to make ideas in the fog.