[zamfi]

Every domain — every language — has its basic jargon, and algebra is no exception. I’m not suggesting that mathematicians simply replace every symbol with a word that hints at meaning; that’s too literal an interpretation of what I wrote. “exponentiate” is no more descriptive than “^”.

But there are alternative ways of describing that derivation that are not as symbol-manipulation heavy; you would certainly not communicate this proof in words to another mathematician — or to a non-mathematician — by simply reading your derivation (or your alternate, verbose derivation) symbol by symbol. Instead you would more likely rely on the meanings of the symbols.

Even in your verbose derivation, however, it’s worth noting that your new names capture the fact that b^x and log_b(x) are inverses, a key piece that someone unfamiliar with logarithms and their relationship to exponentiation now has a hope to understand.

[rocqua]

I think most people who wanted to communicate this proof in person would say 'let's find a whiteboard, or do you have some paper'? And then proceed to write down the first version of the proof.

Because it is better communicated in notation than in words.