Here’s a question to you teachers: How would you teach students to solve equations like

e^{3x} = 0.4 ?

I can think of at least three different ways to explain it to students.

**Method 1.** Use the definition of the logarithm.

log_{ b} x = p is an equivalent statement to b^{p} = x.

So that means the equation e^{3x} = 0.4 is equivalent to 3*x* = log_{ e} 0.4. Divide by 3 to finish solving for *x*…

**Method 2.** Take the natural logarithm of both sides of the equation.

ln e^{3x = ln 0.4}

Then use the power property of logarithms (log x^{y} = y log x) to get

3x ln e = ln 0.4.

Next, remember that ln e = 1, and divide both sides by 3.

**Method 3.** Take the logarithm of both sides of the equation as before. Use a logarithm with the same base as the base in the problem. Then, think of ln and exp as inverse functions so that ln e^{3x} = 3*x* = ln 0.4.

If you say “Teach all three methods!”, consider which one you would teach first. If you only had time to go through one explanation, which one would you use? Is there a better explanation that I’m overlooking?

This question becomes even more nuanced if you have an equation like 5^{3x} = 0.4, and you are interested in getting a numerical answer. In this case, if you use Method 1 or Method 3, then you end up with log _{5} 0.4 in your answer. Since most calculators only give the natural logarithm and base-10 logarithm, one would then need to apply the change of base formula to get a numerical answer. Method 2 might then be slightly less complicated as you could apply the natural logarithm or base-10 logarithm to the problem, regardless of the base present in the equation, and the final answer will be ready for use in a calculator.

Because of this reason and the fact we just got through learning about the properties of the logarithm (like the power property), I opted to show students Method 2 first. This method gets them to practice using the power property and to remember that log _{b} b = 1. I don’t think my students know about the idea of inverse functions so I’m probably going to skip Method 3 (even though this is the way that explain it in my college classes). If I have time, I’ll make sure to explain Method 1. Even though Method 1 is the most elegant, I’ve found that students have a really hard time going between log_{ b} x = p and b^{p} = x, even if you have them write this out before trying to reshuffle the three numbers. If the number x is a complicated expression instead of a simple number, that adds adds to the cognitive demand.

I am raising this issue here not just because I think it’s an interesting question but also because I find that teaching relies so heavily on this special kind of mathematical knowledge for teaching (MKT). Think about all the knowledge that one needs to tackle this question: one needs to know what misconceptions students have about the logarithm, one needs to know what kinds of mistakes students frequently make, what they find difficult when solving exponential equations. One also needs to know that most calculators only give natural or base-10 logarithms. This information is specific to the work of teaching and is not required to actually do mathematics.

There has been some effort to describe and catalog this special kind of mathematical knowledge for teaching elementary school mathematics, but there is little effort at the second school level. I recently attended the National Council of Teachers of Mathematics Annual Meeting in San Diego and I didn’t see any talks on this subject. Why is this the case, if I am right that this kind of knowledge is so important to the work of teaching?

I’ve actually had success using a mix of methods 1 and 3 with high school students this year. The way I introduce logarithms is by analogy with square roots, which they should have seen and be somewhat comfortable with by the time we get to logs. I ask them what it means to take the square root of a number, and what they usually end up deciding (with some guiding questions) is that when you take the square root of something, what you get out is another number that is exactly what you need so that when you square it you get back the original number. I make sure to stress that this is how we defined square roots.

Then, by analogy, I tell them that taking the log base b of something gives you a number that is exactly what you need so that when you raise b to that number you get back your original number. This seems like method 1 to me, and they can usually use this to solve easy log problems.

The next thing I do is continue the analogy. I don’t use the word inverse, but instead I remind them that squaring a number and square rooting “undo” each other. And in the same way, I tell them that taking the log and exponentiating “undo” each other. They usually respond well to this method, but I can see how they would be intimidated if I actually used the language of inverse functions. But with this

Just my two cents.

When you first asked the question, I thought one of the options would be to think about the graph. Before doing any algebra, I think about

y=e^(3x) and then think about y=0.4 and where/how they might intersect. This is like estimating an answer using round numbers, but in the visual sense.

I always taught method 2 and it still makes sense to me as a reasonable way to go about it. I love it that B Fogelson (hi B!) used the other 2 methods.

Regarding MKT,

I blogger I read just posted a review from NCSM of the talk “Learning to do Mathematics as a Teacher” from Deborah Loewenberg Ball. He links to her slides, but looking through them they do look Elementary-centric. I guess we’ve figured out that non-majors need help teaching math but haven’t admitted that math majors might need help too. 🙂

http://blog.mrmeyer.com/?p=6653

I’m surprised that NCSM and NCTM are not more involved with MKT. The physics teachers professional association is very focused on HOW to teach physics — what language to use, what to introduce first, common and uncommon student misconceptions.

And, wow! You’re almost done with the year!

I think in my experience,using the tools for solving logarithmic equations play a bit like “magic flim-flammery” by kids. I find it helpful to have kids verbalize the steps, because a lot of misconceptions emerge:

“Multiply both sides by ln” (is ln a number?)

“After that you can bring the exponent down” (Do they really get why this is ‘ok?’ Is it always ok in solving algebra equations? )

“so ln e of e^3x is 3x” (do they recognize that log BASE e of e^3x involves a inverse relationship?)

I find that pressing to refine / revise their wording pushes the more fundamental issues.

I find that when students truly see these subtleties, they seem to develop better abilities to understand the properties of logarithms (Log(MN)= Log M+log N, log M^k=k*log M )that don’t behave like the algebraic expressions.

They start to recognize that applying a base ten or base e logarithm to the expressions on both sides of an equation allows them to recognize that there’s little need to rely on the “base change” formula.

I also find it very helpful to develop an intuition for log properties by looking at examples with powers of 2, 3, 10. This prevents the temptation to memorize without understanding them. This may be especially helpful if precise verbalizations are tough with ELL’s.

I’m not sure this answers your questions, but I know some early work I do with logarithms allows kids to follow, apply, and understand the different methods when solving equations.