Here’s a question to you teachers: How would you teach students to solve equations like
e3x = 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 bp = x.
So that means the equation e3x = 0.4 is equivalent to 3x = 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 e3x = ln 0.4
Then use the power property of logarithms (log xy = 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 e3x = 3x = 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 53x = 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 bp = 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?