Least understood bit of basic math

Logarithms may be the least understood topic in basic math. In my experience, if an otherwise math-savvy person is missing something elementary, it’s usually logarithms.

For example, I have had conversations with people with advanced technical degrees where I’ve had to explain that logs in all bases are proportional to each other. For example, if one thing is proportional to the natural log of another, the former is also proportional to the log base 10 or log base anything else of the latter [1].

I’ve also noticed that quite often there’s a question on the front page of math.stackexchange of the form “How do I solve …” and the solution is invariably “take logarithms of both sides.” This seems to be a secret technique.

I suspect that more people understood logarithms when they had to use slide rules. A slide rule is essentially two sticks with log-scale markings. By moving one relative to the other, you’re adding lengths, which means adding logs, which does multiplication. If you do that for a while, it seems you’d have to get a feel for logs.

Log tables also make logs more tangible. At first it seems there’s no skill required to use a  table, but you often have to exercise a little bit of understanding. Because of the limitations of space, tables can’t be big enough to let you directly look up everything. You have to learn how to handle orders of magnitude and how to interpolate.

If the first time you see logs is when it’s time to learn to differentiate them, you have to learn two things at once. And that’s too much for many students. They make mistakes, such as assuming logs are linear functions, that they would not make if they had an intuitive feel for what they’re working with.

Maybe schools could have a retro math week each year where students can’t use calculators and have to use log tables and slide rules. I don’t think it would do as much good to just make tables or slide rules a topic in the curriculum. It’s when you have to use these things to accomplish something else, when they are not simply an isolated forgettable topic of their own, that the ideas sink in.

Related posts:

[1] That is, loga(x) = loga(b) logb(x). This says loga(b) is the proportionality constant for converting between logs in base a and b. To prove the equation, raise a to the power of both sides.

To memorize this equation, notice the up-and-down pattern of the bases and arguments: a up to x = a up to b down to b up to x. The right side squeezes an up and down b in between a and x.

 

23 thoughts on “Least understood bit of basic math

  1. This accords with my experience. My younger brother is generally quite adept at math, but he was hardly ever taught logarithms, and had significant difficulty with them when they finally were taught. I tried to explain them to him by reminding him of the concept of inverse functions and then by noting that logarithms are the inverse function of exponentiation — I thought it was the simplest way to teach it, by attaching it to something he already knows well, but I’m not sure it worked.

  2. They’re probably increasingly hard to come by but having one of my dad’s old slide rules (a neat circular one) to play with helped as a tangible, manipulable example of how logs work.

  3. My experience in teaching calculus to undergraduates was that very few understood logarithms. While some were able to mechanistically manipulate equations with log rules, I rarely got the impression that they know what was going on. Actually, I’m not sure that many early undergrads have a grasp on inverse functions in general.

  4. I really don’t think you need to go all retro. In fact, using a graphing calculator is a great way to get a handle on logs. You can compare different bases, compare tables and see the relationship between exponential and logarithmic. I think the real problem is that logs combine inverses, function notation and exponentation. Not to mention that it is invariably the last function introduced in pre-calculus and sometimes not introduced until Calc 2. I remember my calc 2 teacher introducing e as though it was new material, and I’m sure to some, it was.

  5. I think it’s possible to get a handle on logs using a graphing calculator, but it’s hard to avoid getting a handle on them when you use log tables and slide rules.

  6. Yes, people have asked me how to solve that equation many times, usually when they are in chemistry or finance class. I don’t remember really learning about logs until calculus either. Strange that logs have gone out of fashion in primary education, even though their utility has not waned.

  7. Logarithms would be useful in many areas if more people were familiar with them.

    For instance, it would be much better to report market value changes in a logarithmic scale, rather than using percentage. Common mistakes would be avoided, such as believing that an increase by 15% followed by a decrease by 15% brings you back to the starting point.

    In general, any quantities that are more often subject to multiplication would benefit from working in a logarithmic scale.

  8. You’re probably right with logs. For stats experience with standard deviation tables help make things make more sense similar to log tables. After a few questions you start to realize one sided value ~= two sided, when combining two distributions whether it is the one sided or two sided you need and graphically what the looks like. Intuition goes a long way after a bit of experience.

  9. Additive versus Proportional. I think many people have a block when it comes to thinking about proportions and ratios, and are far more comfortable with adding and subtracting. It might take take me an additional 20 minutes more to get to work in heavy traffic, but a more general description is that it takes twice as long from where ever I start.

  10. I used slide rules and log tables around age 12 or 13, and what’s more, I figured out how they worked. I recall wondering why there were no linearly-marked slide rules for adding and subtracting, but in retrospect it’s about as fast to add a couple of 3-digit numbers as it would be to use a linear slide rule, and furthermore, adding the digits gives an exact answer. Perhaps that’s an aspect of slide rules that people wouldn’t get if there were linear ones for adding, that the answer is approximate. I could imagine people using such a thing for balancing a bank account, and wondering why things don’t quite work out.

    Thinkgeek sold a “new” slide rule a few years ago, but I don’t see it available now. Apparently they didn’t sell enough to keep making them.

  11. Among such geeky company (meant as a compliment), I’m embarrassed to admit that I’ve never managed to make sense of slide rules – mainly because the ones I’ve tried to figure out had too many scales.

    There’s an introduction I especially liked in Mathematician’s Delight, by W.W. Sawyer. I wrote about it here.

  12. John,

    I wrote a post last year in which I began with the integral definition of the natural logarithm and proved all of the basic properties of logarithms using only basic calculus. For some reason, calculus texts used by advanced high school or beginning college students do not cover logs in this much detail. In fact, I had to reference introductory analysis texts for some of the proofs.

    My post can be found here: http://thinkingmachineblog.net/logarithmic-and-exponential-functions-for-high-school-advanced-placement-and-university-introductory-calculus-part-1-logarithmic-functions-of-real-variables/

  13. Many people consider calculus to be a difficult subject, but I don’t think that’s actually true. I think it’s more that it places a heavier burden on mastery of preparatory mathematics, such as trig and algebra, and the encounter with calculus is more likely to reveal the generally shoddy state of math education that most students get these days. This point about logarithms is a perfect example of this.

  14. “To memorize this equation, notice the up-and-down pattern of the bases and arguments: a up to x = a up to b down to b up to x. The right side squeezes an up and down b in between a and x.”

    Eons ago my math teacher thought another similar rule: think of it as a fraction where base and argument are denominator and numerator. That gives: x/a = x/b * b/a

  15. Michael Herrington

    John,

    Introduced logs to my daughter at the end of last school year. They will come up again this year. I spent a lot of years teaching math, but most of it at the middle school level, so logs for me are rusty as well. Thoughts on best approach for logs. I have an old slide rule, but its use is rather foreign to me.

  16. Maybe what makes logs difficult is the idea of an inverse function. Maybe start there: clockwise turns are the inverse of counterclockwise turns, subtraction is the inverse of addition, etc. Then move to logs base 10. 10^2 = 100, so log_10 100 = 2, etc. Then point out that there’s nothing special about base 10. Could do the same base 2, which is very convenient in computer science. Then mention that bases don’t have to be integers. One could take logs base pi, for example. Or logs base sqrt(2), because there you could come up with some examples that work out nicely.

  17. One important thing about log that many people seem to forget: It is monotonic. If your original function is monotonic, then the log of that function will be as well. Quite handy.

  18. Indeed. For example, in statistics, you usually maximize the log of the likelihood function, seldom the likelihood function itself.

  19. I graduated high school in 2009 and we had a giant 6-foot slide rule on wheels which was used for a few lectures in my high school math classes.

    Radiolab did a program about numbers a while back, and the first story was on how humans are born to do logarithms, but we beat the logarithms out of people when we teach them natural numbers. The other two stories are good, too. http://www.radiolab.org/story/91697-numbers/

  20. Maybe the metaphor of the “width” of a number would make it easy to teach logarithms?

    (okay, not really a metaphor — that’s pretty much what logarithms are for positional number systems.)

  21. imho it’s a bit antiquated to talk of log as the least understood but most useful bit of elementary math.
    now we have fourier-transform and fft, and it’s well understood by mathematics. but not by others.
    also generally infinite-dimensional linear algebra is rarely understood well.
    seems at my uni, the one who understands these things best also happens to have international success.
    so on the bachelor level, imho functional analysis is the least understood topic.
    and using fourier-transform could help in many areas and not just in sound and video processing.
    I would strongly suggest to teach fourier-transform thoroughly in middle-school, maybe instead of log.
    of course I’m saying that only because Computer Algebra Systems usually are better at fourier-transform than at doing algebra with log-functions…

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