Logarithms

A young teacher writes:

I’m teaching exponential functions and just getting into logs. Everyone tells me that kids tend to struggle with this chapter, so I’d like to try something different than what my school has been doing (just teaching the rules, then practicing over and over again.) Is there a way to introduce logs that makes them seem a bit more connected to exponential functions? We’re going to talk about how they’re inverses, but I think what my students get stuck on the most is notation, so I’m worried that logs will freak them out. Do you have any ideas for how to practice going back and forth between log and exponential form, and for introducing rules of simplifying and expanding logs?

Students indeed freak out when logs are introduced. Part of the reason is that the introduction is often done in a way that gives the concept no grounding in student understanding. Even an eloquent explanation by a brilliant teacher does not help much, because it does not answer a question the students have, and most of them just cannot hear it.

As with other intimidating new ideas, I think it is best to base it on something the students already understand, and then, when it starts to take root, introduce the terminology and the notation. That is what I do with the trig ratios: I start by discussing the slope that corresponds to a given angle with the x-axis, develop an introductory feel for it through measurement, and use those measurements to solve some “word problems” involving right triangle trig. By then, students are using trigonometry without knowing it, and I can introduce the tangent ratio by name, as well as the fact that it is available instantly on the calculator. (A version of this approach to basic trig can be found here, and in my Geometry Labs book — free download.)

A similar approach to logs is what I call super-scientific notation. The prerequisites are an understanding of scientific notation and the laws of exponents. This approach requires access to a tech tool for graphing or equation-solving. The idea is to tell students that any number can be written as a power of 10. For example, take the number 2300. That is 2.3 · 10^3 in scientific notation, but if we want to write it simply as a power of 10 (super-scientific notation,) the exponent must be a number greater than 3, since 2300 > 10^3, and less than 4, since 2300 < 10^4. To find what that power must be, we can graph both y = 2300 and y = 10^x, and find their intersection.

(See TI-eighty-something screens leading up to this here.)

Of course, the solution suggested by the graph (3.36) is approximately the log base 10 of 2300, but I do not use the L word yet. Instead, I have the students find the exponents in super-scientific notation for many numbers, and then make no-calculator calculations using those. (Download the worksheet.) That lays the foundation for the laws of logs.

This activity is not sufficient alone to take the sting out of logs. Indeed, once the term and the notation are introduced, there is usually a setback, but the freak-out is less than it would have been otherwise. In fact, the activity provides an excellent anchor to the unit. Super-scientific notation can be referred to repeatedly in the following days, whenever students are confused or make mistakes in applying the laws of logs. Reminding them of this introduction is an excellent way to stress the relationship between logs and exponents.

A related activity, Make Your Own Slide Rule, helps create a visual representation of logs, and introduces students to a bit of history: how logs were used before the advent of the calculator.

–Henri

PS: an interesting, complementary approach can be found in a blog post by Anna Blinstein.