# The "rearranging" approach to teaching logarithms

Consider the following way to teach division:

Division works this way: any product equation $$xy = z$$ can be rewritten as a quotient equation $$x = \frac{z}{y}$$. Just move the numbers in that way.

This is true, but is a horrible way to think about division! Instead we show students that they can divide both sides of an equation by y, and then that multiplication is the inverse operation of division so the two resulting y's on the left cancel.

Consider the following way to teach logarithms (which is very common):

Logarithms work this way: any exponential equation $$x^y = z$$ can be rewritten as a logarithmic equation $$\log_x z = y$$. Just move the numbers in that way.

This is true, but isn't it equally bad? Shouldn't we be demanding that students take the $$\log_x$$ of both sides?

Many algebra textbooks contain questions as follows:

Convert the following logarithmic equation into an exponential equation: $$\log_5 x = 2$$.

They imply that the way to deal with logarithms is to move symbols around. Is there some [historical? pedagogical?) reason this approach is so widely accepted even though the same approach seems ridiculous when talking about multiplication and division?

• The logarithm is usually defined by moving symbols around. Alternatives like power series or differential equations are inaccessible to students of that learning stage. (German curriculum.) So, unless you know, what $\log_x$ means by moving symbols around, you can't understand what taking $\log_x$ of both sides means. Then of course, you should prefer taking $\log_x$ resp. $x^{()}$ of both sides to work with. Apr 10, 2014 at 17:43
• @DaveLRenfro, why not just introduce the logarithm as the inverse function of the "power to the base $e$" function? I learned them that way (base 10, before calculators) in school... Apr 10, 2014 at 18:29
• @vonbrand Besides: Symbolically, there is a triangle of inverse operations: a power ($a^b=c$), a root ($a=\sqrt[b]{c}$) and a logarithm ($b=\log_ac$). Functionally, there are two pairs of inverse function: Power functions vs. Root functions and Exponential functions vs. logarithms. But the connection between these two is difficult to understand. Apr 10, 2014 at 19:17
• @vonbrand: In thinking about this, I think (it's been nearly a decade since I've taught) at least a few times the very first thing I've done was to write in a vertical column on the board $3^2 = 9,$ $4^2 = 16,$ $5^2 = 25,$ etc., then in an adjacent vertical column $3 = \sqrt{9},$ $4 = \sqrt{16},$ $5 = \sqrt{25},$ etc. (Maybe even explicitly showing the $2$ root index.) Then further over I'd write the exponential column analogs: $10^3 = 1000,$ $10^4 = 10000,$ $10^5 = 100000,$ etc. and $3 = \log{1000},$ $4 = \log{10000},$ $5 = \log{100000},$ etc. Apr 10, 2014 at 19:21
• @Toscho I don't understand why you posted your answer(s) to the question as a comment?? Why not in an answer? Apr 10, 2014 at 19:23

I think the approach to division (and logarithms) that you complain about is actually the right one. It seems to me that the statement

$$x = \frac{z}{y}\qquad\text{is the same as}\qquad xy=z.$$

is essentially the definition of division. For example, if you were to ask a third grader the question

Is it true that 12 divided by 4 is 3? If so, why?

you would hope to get the answer

Yes, since 3 times 4 is 12.

This would be much better than the answer

Yes, since we can divide both sides of the equation $3\times 4 = 12$ by 4 to get the equation $3 = 12 \div 4$.

In the same way, the proper answer to the question

Is it true that $\log_{10}(100) = 2$?

is

Yes, since $10^2 = 100$.

A student who has a good understanding of logarithms should be able to do this in one step in their minds, without writing down any equations or thinking about applying operations to both sides.

Now, you complain that this approach involves "moving symbols around", but I counter that the symbols in mathematics have meaning. In this case, the way that we're moving the symbols around directly reflects how we think about logarithms.

Finally, I should point out that this really just has to do with two different definitions of inverse functions. Here they are:

Definition 1. Two functions $f\colon X\to Y$ and $g\colon Y\to X$ are inverses if $$f(x) = y\qquad\Leftrightarrow\qquad g(y) = x$$ for all $x\in X$ and $y\in Y$.

Definition 2. Two functions $f\colon X\to Y$ and $g\colon Y\to X$ are inverses if $$g(f(x)) = x\qquad\text{and}\qquad f(g(y)) = y$$ for all $x\in X$ and $y\in Y$.

It seems to me that you prefer using definition 2, but I think definition 1 is simpler.

• "$x=z/y$ is the same as $z=xy$" Almost. I can definitely say that $0\times 0=0$, but writing $0=0/0$ makes me somewhat uncomfortable, to say the least. Teaching "inverse operations" with full understanding when the inversion is possible and when it isn't looks completely reasonable to me, but just declaring that "one can get an equivalent equation by moving the symbols around" is a bit fishy. Jan 16 at 3:43

Being able to work out how to convert between exponential and logarithmic equations can practice a really essential algebra skill: what to do if you almost remember a rule.

I instruct students who forget how to convert between exponential and logarithmic equations to simply remember one true logarithm fact. (This is aided by the fact that I reviewed logs base 10 previously, so they can remember by counting zeroes that $$\log_{10}(1000) = 3$$.)

They then write the fact down and the corresponding true exponential fact $$10^3 = 1000$$, then replace all the numbers with letters to get a conversion formula $$\log_b(c) = a \iff b^a=c$$. Then they plug in whatever they want for $$a,b,c$$ to get a new true equation.

While this trick won't let students generate new algebra knowledge, and requires a bit of care to remember a good example ($$100 \cdot 100 = 10,000$$ won't help you remember if the rule is $$x^a \cdot x^b = x^{a+b}$$ or $$=x^{ab}$$), it can help them prepare in advance to rederive rules from class, or identify if rules are wrong before using them. (I'd love to have a grading system that rewards students if they:

• Identify that they have misremembered a rule
• Demonstrate by plugging in an example that the remembered the rule wrong
• Solve the problem using the wrong rule as if it were right

)

That being said, I then also show students the "take logs or exponentials of both sides" approach to converting log and exponential equations, which I find a more appealing approach, since it gets students thinking that taking logs or exponentials of both sides of an equation is a helpful thing to do.

A side note about teaching logarithms:

My understanding (and please correct me if I'm wrong) is that it used to be that students would encounter logarithms earlier as tools for multiplication in arithmetic, but students nowadays may be seeing logarithms for the first time in an algebra (College Algebra) class.

It's a lot to throw at students all at once:

• Function notation
• Subscripts
• Exponent rules
• Functions
• Inverse functions
• A brand new arithmetic operation they don't have an algorithm for computing (they don't have an algorithm for square root either, just intuition, so this can be a good way to explain how to think about logs)
• Applications of logarithms
• Solving logarithmic/exponential equations

So I suggest spreading these topics out over the semester/between semesters and not trying to cover all of them in three class meetings.

I'm curious where those "quoted" parts came from. It's not about "division works this way" or "logs work this way", exactly. Those are definitions. When kids first learn division, they aren't ready to hear about definitions. But when we teach logarithms, I hope that most students are ready to hear about definitions (and proofs).

I definitely teach students to work with logarithms in the same way they do most other algebra steps: do the same thing to both sides.

I think there are some problems for which it helps to refer back to the definition. (But I'm not thinking of what they are at the moment. I will come back and add to this when I'm able to.)