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?

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    $\begingroup$ @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... $\endgroup$ – vonbrand Apr 10 '14 at 18:29
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    $\begingroup$ @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. $\endgroup$ – Toscho Apr 10 '14 at 19:17
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    $\begingroup$ @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. $\endgroup$ – Dave L Renfro Apr 10 '14 at 19:21
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    $\begingroup$ @Toscho I don't understand why you posted your answer(s) to the question as a comment?? Why not in an answer? $\endgroup$ – Chris Cunningham Apr 10 '14 at 19:23
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    $\begingroup$ @Dave L Renfro Hi again! Why not post an answer instead of these comments that answer the question? :) $\endgroup$ – Chris Cunningham Apr 10 '14 at 19:24

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 $\log_{10}(100) = 2$?


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.

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