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Friday, December 18, 2009

Introducing Logs

This is nothing earth-shattering, but I feel my Algebra 2 students are less freaked out by logarithms this year, and I think it has to do with how I first introduced them. I used to start by declaring A LOGARITHM IS AN EXPONENT, like saying it loudly and slowly would help it sink in better. I should really know better by now. Well ok maybe, I do know better by now, because this year I started by inviting them to play a fun puzzle:
(Adapted from James Tanton's monthly St. Mark's Math Institute newsletter.)

And then let them in on the dirty little trick that in math we insist on calling it a logarithm instead of a power when we write it like that.

To answer the inevitable "What the hell?", I go into a little history of how John Napier invented logarithms to make multi-digit multiplication easier for Renaissance astronomers.

Then we mention that it's an inverse of an exponential equation, play a little more with shifting between exponential equations and log equations, and we are done for the day.

Here are my filled in notes:

Here is the smart notebook file. Here is a google doc with text of a task for easy copy and pasting.


  1. So what's with every time I hear about credit card numbers or even the way google searches for things, I always hear about some complex logarithm. I never thought it had to do with exponents, I thought it seemed like some shortened method to tally things up or something.

  2. I think the word you keep hearing is "algorithm." Sounds kind of the same but TOTALLY DIFFERENT WORD. An algorithm is basically "a procedure" that will always give you the right answer no matter the inputs. Like the algorithm for dividing by a fraction is "keep change flip." and the algorithm for subtracting an integer is "add the opposite sign." Computers are very good at following algorithms because they can follow a precise procedure a brazillion times a second on whatever inputs you throw at them. That's why you are always hearing about google's search algorithm, etc.

  3. It's interesting how that parallels my own change in the way I teach logs. A few years ago, another teacher and I discussed it and decided to teach the exp() function in the same way that you are using the power() function here. It worked ok, but I eventually moved to a symbolic representation of logs, instead of a text representation of powers. Now my students learn it even better than before. Unfortunately, I have to wean them off of my homemade log symbol before they leave my class. Here are a couple of my posts on this topic:
    The Big L
    Intro to Logarithms

  4. Oh Dan I do remember the Big L notation but I forgot I knew about it. Hm. Works better you say? Then I guess I'd just have to show them how to convert log() to big L before the regents. Thanks for reminding me about that.

  5. My students, in general, need more scaffolding than most, and the Big L works better for them. (When I tell them that Big L is something I made up, and not how they will see it outside of my class, they say I should "tell them to change it".)

    But does it work better in general? I have no idea - no one else has tried it (that I know of). Would showing your students a new notation now be helpful or confusing to them? Mine always get upset when I try to show them too many ways to do something (i.e. more than one), but that's a function of their fragile sense of self-confidence in math.

    It might be interesting to show it to your students and see what they think about it. I think it might have even more benefit when working with log properties, as it makes them much simpler to write and remember.

  6. I'm stealing this.

  7. This is great! Here's something else I'm going to reuse from you. Such genius, going from telling students to making them do it... So simple and elegant!

  8. Yeah Dan I haven't decided about this year. I think even just showing them a different notation might just frustrate and confuse them. I was thinking for next year, just act like big L is the accepted notation right off the bat. Something like that really would be more consistent with the rest of the notation we use than the log_b(). It's funny how we don't even think about things like that because we've always done it that way. Well, I don't. You do.

  9. This reminds me of something I saw years ago called "Super Scientific Notation"

    though I like your idea better

  10. Kate, I just wanted to let you know I've been reading along and enjoying your blog for a while. This post about logs is SO AWESOME!!! I am really excited to try it with my students.

    My students and I figured out another way how to remember which part of the logarithm is what -- "base, answer [in subscript], equals exponent." The A ("answer") goes next to the B ("base") and the exponent goes next to the equals sign (Es go together).

    One student of mine who struggled to remember basically anything was able to remember how to rewrite logarithms as exponents and vice versa using this method.

    But your way will probably work even better.

  11. Steve, Rebecca, thanks for sharing. Rebecca your method looks familiar - I might have been taught that way myself. Whatever works!

  12. Just today I introduced logarithms this way and I must credit your ideas for helping build an excellent first day. The "puzzle" start was perfect (especially with a few students weary of the intimidating word "logarithm" written in my objective), and I really felt like introducing the notation with "power" first made it more familiar and inviting, and it fit well with how our department sets this unit up. Thanks for the great tip (and keep up the excellent blog!)

  13. @ Mr. Condescending
    @ Kate Nowak

    While it's true that algorithms sound a lot like logarithms and can be misinterpreted, I think Mr.C might have actually heard the word Logarithm in reference to his credit card example. Cryptography (the stuff that makes sure nobody can steal your credit card information when you buy online) is often based on the number theoretic concept of 'indices' which are actually very similar to logarithms but work in modular arithmetic. Someone talking about credit card security might refer to logarithms.

  14. First of all, I should confess that I probably love this because I too like to know up front who in the history of mathematics I should blame for things that trip me or my students up. :)

    Then, once I get over that*, I can appreciate the subtle brilliance of a well-crafted framework.

    - Elizabeth (aka @cheesemonkeysf on Twitter)

    *Of course, sometimes this takes me a few years.

  15. Kids can figure out what a logarithm is on their own. I've done something very similar w/ my students. I present the puzzle using "log" rather than "power" and the puzzle is to figure out what this means. Like you, I start w/ some simple ones such as log 2 8 = 3 and work up to negative and rational exponents. My last question is something like log 7 7^6. Having students communicate what is being asked is very helpful. I have some students actually calculate 7^6 first and then figure out what power of 7 this large number is. Kids who wrote or verbalized what was being asked ("What power of 7 is 7 to the 6?") are the ones who realize how simple this question is.

  16. Oops. Reading my comment above, it looks different than it sounded in my head. Should be "Kids can figure out what a logarithm is on their own." When we believe in them and let them do the math, as Kate has done here, they will figure it out (and feel more confident about themselves as mathematical thinkers in the process).

  17. Thank you for sharing your nifty idea but I would suggest a variation that shies from giving away the property through its name (power). So, give the property any name you see fit other than the word power itself (use the corresponding name for power in a different language for example.) This, hopefully, would make the problem both a deciphering problem as well as a discovery of logs problem.


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