The Natural Log

The calculator has two built-in log functions: log-base-ten, denoted by “log” (no subscript), and log-base-e, denoted by “ln” (evidently named by a Frenchman; loh-gah-reeth’m not-chore-ell, with an adjective postpositive)—the natural log. Logs to the base ten are obviously useful—it’s not by accident that I gave an example of the “power-to-product principle” (as I here propose to call it until something better comes along) yesterday using logs base ten. But what’s up with e?

The short answer is “it’s a Calc thing; you wouldn’t understand”. Probably the right answer, too. I don’t know if introducing e before limits is a good idea or not; I do know for sure that that it’s way overstressed in the standard textbook treatments. Once they’ve decided to include natural logs, they use ’em even when another base would be easier. (specifically, in growth-and-decay problems, where the really “natural” choice of base would be “2” [because of “doubling times” and “half-lives”]). This is of course the opposite of mathematics: using bells-and-whistles to obscure the truth.

From master expositor Eli Maor‘s e: The Story of a Number (reviewed here), I recently found out that the “compound interest” approach to explaining the role of e isn’t nearly as contrived as I’d assumed. In preparing the last remark, I learned that Maor’s Trigonometric Delights is available free online from U of Princeton press. This is just amazingly cool. And it’ll even turn out to have something to do with e.

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