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.
Math 148: Precalculus 
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?