Subnetting (read "Logarithmic") Arithmetic

All networking types daily make use of arithmetical operations optimized to deal with a fixed number of digits in base-2. These operations are most often performed mentally, with little thought given to the underlying rules the optimized manipulations obey. This is not to say the mechanics of the operations are not considered--indeed, it is only by acute awareness of the binary structure that one may come to grok subnetting at all--but rather the leap from binary arithmetic to subnetting arithmetic is often only tacitly understood. However, in tutoring others on this topic, I've come to develop what I gather is a very clear picture of how we optimize our day-to-day calculations. And, most importantly, my insights are universal in that many others have independently arrived at the same conclusions I have. Therefore, I believe my attempts at developing rigorous notions of these optimizations can be of some value.