exp & log

Bad news first: in the pros, “ell-oh-gee” spells natural log; the symbol ln has been foisted upon generations of students by a satanic pedagogy. One guiding principle for this amazingly widespread ideology is that a sticking point—whatever tends to stump the students the most—having been once identified, should be preserved so that the student will be confused for as long as possible.

The honest solution is to admit that in some lectures, or chapters or what have you, “log” will mean “log base ten”, while in others it’ll mean “log base e“, and that it’ll seldom mean any other thing than one of these, but if it does then obviously it’ll still be a log function and it’ll probably be pretty obvious from its context what log function. Or it’ll be a generic log function, in just the same way that f typically denotes a generic (real-valued real) function; it’s hardly that weird of an idea after all. This would spare instructors from writing out subscripts and diligent students from copying them in making statements like log_a(x/y) = log_a(x) – log_a(y); the easy thing is to set up appropriate definitions at the beginning of a given discourse; part of “appropriate” is that frequently-used symbols should be easy to write down.

I’ll follow the textbook conventions here—log means log_10 and ln means log_e but intellectual honesty compels me to admit that when you see ell-en, I usually say “log”. Oh, and exp doesn’t mean anything in the world but exponential-base-e (and has little to do with the rest of the post).

So the first thing is that examples oughta be given in base ten liberally. And that somehow I’ve mostly done it sparingly so far. Another one of those “ideally, this would be an exercise” things we like to toss off while we grind out whatever horrible little lesson plan we’ve tied both hands to on the given day. But anyway, I’ve done some of that here… that’s part of how I know I’ve done too little of it in class. So what else?

I jumped ahead for the change-of-base formulas first thing this quarter; what’s the rumpus?Well, a gut feeling that changing bases feels like it’s pretty close the center of at least one of the sticking points that the industry seems to wish to preserve.

In the first place, it’ll be easy once you see it. Nobody believes me when I say things like this (and no wonder… whenever anybody says things are easy to me, I figure they’re probably lying…) but it’s the exact kind of thing I oughta be an expert in by now and anyway I’ve been paying attention, so maybe my judgements (and pronouncements) on such matters should be treated as at least worthy of consideration…

In the second place, translating everything into terms of e—in a Pre -calc course, seems sort of horribly misguided. The logs-and-exponentials that matter in the introductory lectures should be (certainly) ten and (almost as certainly) two (and I’m leaning strongly to three as the number one contender for third place).

Base Ten for setting up the (very proof-like) mnemonic devices I’ve already been discussing—one uses log(1000/10) = log(1000) -log(10) to illustrate a more general fact; this kind of thing is how the Faithful avoid “memorizing” “rules”.

Base Two for the graphs. And for many other good reasons (quick, pick a positive number bigger than one; betcha didn’t say two-point-seven). The only reason for leaving the easiest possible case out of a lecture series would be to leave it as an exercise… but that’s a whole different level of the game.

And once we’ve got two bases, how they relate one-to-another becomes a perfectly natural question, particularly when put in the form: “well, right here’s the ‘log’ key; how the heck do I get ‘log-base-two’?”. And that gives me an excuse to say: by transformations! Just like at the beginning of the course! Later.

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