John Napier, the inventor of logarithms, allegedly got his last name when it was said of him (perhaps by himself) that he “hath nah peer.” I hope to convince you that, for back-of-the-envelope thinking, logarithms and logarithmic thinking have no peer.
The fundamental reason is that multiplication is more useful than addition. Here is the proof by example.1 Imagine that you have to add two numerical contributions—perhaps two energies, E1 and E2. The possibilities for their sum are threefold. If E1 ≫ E2 (the first effect dominates), their sum is just E1. If E1 ≪ E2, the sum is just E2. Finally, if E1 ∼ E2, the sum is just 2E1 (or 2E2). Thus, forget addition and instead use multiplication (here, by 1 or 2).
This principle of estimation...
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