We’re heading into the home stretch: logs and exponentials. I’ve *glanced* at the exercises of the type “use transformations to graph” such-and-such functions (*y = 3*2^(x-1) + 5* in my actual example [AM only]): the point here isn’t that they’re not important—this could potentially be some of the most valuable experience a student gets around here since we’re simultaneously “reviewing” material that may very well not have been perfectly clear, while *at the same time* becoming at least a little bit more familiar with the properties of “log” and “exp” functions.

No, the point for me is to *get the calculator involved*. So, without slowing down to prove ’em, I’ve cited the mighty fact (which, along with the still mightier “A Log Is An Exponent”, I hear in my teacher’s voice: thanks, Greg Peters): The Log Of A Product is The Sum Of The Logs. Also *log_a (x^y) = y*log_a (x)* (I don’t have a mantra for this and could probably use one; TeX is broken but you can probably read the code). The key example here is something like *log(10^5) = log(100000) = 5 = 5*log(10)*; we are (“obviously”) using logs-base-*ten* in these equations.

We’ll prove these next week (there’s a day off tomorrow; weirdly they claim to be observing “President’s Day” though obviously they’re doing something else).

Armed with these, I can get down to calculating some logs. Two different ways. Suppose I want to know the log-base-two of 31. Rewrite as an exponential (this would be the *only* move right now if I hadn’t “cheated” and mentioned the yet-unproven facts just cited): *2^x = 31*. “Obviously” *x* is about 5; we can calculate it to, whatever, nine-place accuracy *graphically* by a “102” method: find the *x*-co-ordinate for the intersection of *Y_1 = 2^x* and *Y_2 = 31*. Okay, cool. Maybe logs aren’t so bad if we can calculate ’em so easily.

But even this is too much work. Go back to *2^x = 31*; take ell-oh-gee of both sides (“log” = “log_10”, the so-called “common log”; if any student of mine doesn’t know this *for the rest of their life* beginning next week at the latest, I’d be disappointed to find that out); apply the mantraless fact of a few paragraphs back; out pops *x = log(31)/log(2)*, which one can now punch into the calculator directly to get the same result without mucking around with graphs. Experienced hands will recognize the Change Of Base Formula here; there’ll be more to say of course. Have a good weekend.

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February 27, 2009 at 3:59 am

I’ve never heard of the saying “The Log of the Product is the Sum of the Logs” and to this day I have to think about it to get it right, so I know it’s easy for students to get it backwards. A different saying, one I just made up, thank you very much, might be “Some of the logs are logging products” although this might imply that it’s only true some of the time, I suppose.

Ron

February 28, 2009 at 5:27 pm

pretty quick turn of the phrase, there, ron.

but of course i’ll take every opportunity

to use a quick calculation (like, oh, say,

log(1000) = log(100) + log(10)for example), and repeatedly too,

with pretty much any intro-to-logs setting.

i’ve been saying (mostly to myself)

“trust the code” quite a bit; the phrase

“use the source” (with or without “luke”)

also tends to come to mind. right *now*

this means “verbal mnemonic devices

are all well and good… but don’t let

verbal tails wag mathcode dogs

when things are

simple“:a^(x+y) = (a^x)(a^y)not because this is some memorized fact

but “because” (actually *for the same reason as*)

(x^3)(x^2) = (xxx)(xx) = (xxxxx) = x^5.i’m not above begging if it’ll get students

to write out little proofs like this…

February 28, 2009 at 5:35 pm

not that i won’t be mentioning–

and that right early–

“some of the logs are logging products”…

September 15, 2009 at 7:25 am

saying in math?