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.

March 1, 2009 at 6:31 pm

Binary is so ingrained in my mind that I almost always mean log_2 when I think log. When I write about logarithms I always use the subscript, just to be clear. Sometimes I need log_10, but I don’t think I’ve used log_e since college.

I use change of base to compute log_2, using whatever base log function is available in the language I’m using (there’s not usually a native log_2 function). That’s a practical example of why it’s useful.

March 1, 2009 at 10:47 pm

i read somewhere that computerheads

sometimes refer to

log_2aslg.this reference begins to convince me

that this question is probably best ignored.

rick’s newish blog–exploring binary—

is a beaut. vlorbik sez check it out.

March 2, 2009 at 3:46 am

Yet, again, I wander in from the dark, searching for clarification that only Vlorbik can offer. “ell-oh-gee” and e have been chasing each other all weekend, and I’ve been running to keep up. Finding myself at a loss, I return to The Blog for direction and reassurance. Thank you, Vlorbik, for restoring order, at least for the moment. Although I am beginning to see a faint light at the end of the tunnel, please help me understand; how does the not-yet-initiated determine when to “ell-oh-gee” or, when does e better serve?

March 2, 2009 at 3:59 am

that is “the” solution

to the differential equation

y’ = ymeansamong many other things that the base

eis the simplest exponential function in *calculus*.

for more detail, see, e.g. the maor link a few posts back.

there are quite a few entire passages photocopied

verbatim. this “google books” thing is jawdropping.