The fog on the bathroom mirror is caused by condensation. Falling air temperatures cause condensation, because warm air can hold more moisture than cold air.
When warm air in the bathroom is heated by the shower to temperatures as much as 100 degrees Fahrenheit, the air becomes saturated with moisture. When this saturated hot air comes in contact with the bathroom mirror, which may be as much as 30 degrees Fahrenheit less than the air temperature, the moisture begins to come out of the air and be deposited on the mirror surface. The clouds of steam in your bathroom are basically raining on your mirror.
Installing a mirror heater will keep the temperature of the mirror high enough to prevent condensation.