# Apply the inverse operation on both sides, or know the inverse function?

My old question here was about logarithms, but as I teach more and more (precalculus) algebra, I've generalized the question a bit in my mind.

Should students do the same thing to both sides, or know the inverse function well enough to move symbols around?

I'll try to illustrate what I mean with some simplified examples.

Here are two possible solutions to the question "Solve for $x$: $x + 5 = 12$."

Subtract $5$ from both sides. The $+5$ cancels the $-5$ on the left so we get $x = 12 - 5$. So $x = 7$.

or

Move the $5$ to the right side to get $x = 12 - 5$. So $x = 7$.

In the first case, the student applies the inverse of a one-to-one function to both sides of an equation to get a new equation.

In the second case, the student appeals to the core definition of what subtraction means: the whole idea of subtraction is that if $x + 5 = 12$ then $x = 12 - 5$.

Similarly here are two possible answers to the question "Solve for $x$: $log_5 x = 3$."

Exponentiate both sides with a $5$ to get $5^{log_5 x} = 5^3$. The $5$ and the $log_5$ cancel on the left so we get $x = 5^3$. So $x = 125$.

or

If $log_5 x = 3$, then $x = 5^3$. So $x = 125$.

In the first case, the student applies the inverse of a one-to-one function to both sides of an equation to get a new equation.

In the second case, the student appeals to the core definition of what a logarithm means: the whole idea of the logarithm is that if $log_5 x = 3$ then $x = 5^3$.

Advanced students know that you can "move symbols" in certain ways to rearrange the statement in an equation, so an advanced student is more likely to successfully implement the second strategy. Additionally, it seems to me that math faculty tend to favor the second strategy.

However, novice students are more likely to succeed with the first strategy and are more likely to be implementing the core mathematical plan "do the same thing to both sides" throughout mathematics. Which of the two strategies should we favor, when, and why?

One more example: "Solve for $x$: $(\pi - 1)x = 12$." Would you tell a student to "divide both sides by $\pi - 1$?" Or would you ask them to move the symbols around, hoping that they are correctly appealing to the definition of division? Is your answer different if the objects involved are more or less complicated?

Note: Jim Belk's answer to the question linked at the top is very insightful and I am hoping for more discussion of that type, or at least a reference-request for what is this pedagogical concept called?

• I think that students should be expected to justify their work, or at least be able to provide justification. Applying the inverse to both sides is "self-explanatory". "Moving things around" would probably need to be backed up with something like "by the definition of the logarithm, $\log_2(x) = 8$ means that $x = 2^8$" . May 1 '17 at 18:37

I'd say both: they must know the inverse function, and that it's being applied to both sides. This then connects up with the fundamental properties of equality. As a teacher of many algebra courses (elementary to college level), students who think of "moving" terms around can almost never generalize to the next level.

• So to clarify, you think we should deprecate the logic "$log_2 x = 4$ means $x = 2^4$" and focus more on an intermediate step $2^{log_2 x} = 2^4$? May 1 '17 at 14:59
• @ChrisCunningham: I'd say both. You can't get away from the logic of the former because that's actually the definition of a logarithm. However, the latter appears regularly and I feel the statement is easier to remember and generalize. In practice I would probably solve with the latter and then double-check with the former. May 1 '17 at 15:16
• @DanielR.Collins I wouldn't say that's the definition of a logarithm. All the logarithms I have seen have been defined by either $\ln(x)=\int_1^x \frac{1}{x} dx$ (for x positive) or as the unique function such that $f(e^x)=x$. The latter is somewhat close to the above mentioned but not really there. (of course for different base logs you just define them as $\log_a(b)=\frac{\ln(b)}{\ln(a)}$) In other words the above is a useful way to remember what $\log_a(b)$ is but it's probably not the definition.
– DRF
May 12 '17 at 4:39
• @DRF: The definition I see in all my college algebra, precalculus, and calculus texts, Baby Rudin, Wolfram Mathworld, etc., matches what I wrote above. E.g.: mathworld.wolfram.com/Logarithm.html May 12 '17 at 13:21
• @DanielR.Collins How then do you define $a^b$? Through limits of series? As for the definitions in Wolfram while they do define logarithm as you say. They don't define exponantiation. They define $e^x$ and $e$ using the differential equation and the solution of $\int_1^x \frac{dt}{t}=1$. Which is fair enough. But I can't find a definition for $a^b$ anywhere.
– DRF
May 12 '17 at 13:52

As a practicing mathematician, I don't even distinguish between the two options you describe. You're not presenting two different options, but two descriptions of the same option.

"Moving" the $5$ to the other side (or, more accurately, getting rid of its effect on $x$) is an effect I want to achieve. Subtracting 5 from both sides is the mechanism by which I achieve this effect.

Incidentally, I think I would also use the two descriptions of the same option mostly as as synonyms. So, I would be wary of assuming that people who sound like they speak in terms of one description are making an actual choice between the two.

So, my unqualified* opinion is that you need both — having students recognize what goal they are trying to achieve is just as important as having students recognize what technique to apply to a problem. And, of course, to learn how to discern which techniques can be used to achieve which goals and when.

*: I am not an educator

The flip side of your concerns, I think, are the students that don't develop a goal-oriented approach to problems, so when faced with a problem that doesn't match one of their recipes and so they are lost when they have to develop their own goals for manipulating formula.

Or worse, then the goal for a specific problem is different than the 'usual' goal one would have for the formula; e.g. calculus problems where one wants to manipulate an equation to make $x-1$ appear someplace, rather than the 'usual' goals of simplification.

• I like the point of a goal-oriented approach; this implies to me that you are an educator at least to some degree. But I do believe that when teaching algebra, you need to choose one or the other of the two options (even though from a certain perspective they are the same thing). May 1 '17 at 14:57

The fundamental concept is always: You can do the same thing to both sides, as long as it does not involve an illegal operation (like dividing by zero, or taking the logarithm of a negative number). But there are an infinite number of choices of "things" you can do to both sides, most of which don't help solve the problem. The student needs to know what the relevant inverse operation is, in order to choose which "thing" to do to both sides.

• +1. Can I ask for clarification though? Do you think a student should solve $log_2 x = 4$ by exponentiating both sides, i.e. $2^{log_2 x} = 2^4$? Many people do not think so. May 1 '17 at 14:58
• @ChrisCunningham -- Daniel Collins' comments make sense to me. It has been a long time since I have had to do a lot of "logarithm in arbitrary base" problems, so my memory of "how to teach it" is fuzzy. I also don't have the relatively large sample size of trying to teach many students, so I don't know what approach works for the largest number of students. May 1 '17 at 17:40
• My mother's advice about teaching was that a good teacher knows five ways to explain something, and can come up with a sixth and a seventh for the student(s) who look at things from a completely different angle. May 1 '17 at 17:40
• When I do find myself doing logarithm problems by hand (or in conjunction with a calculator), they tend to be a bunch of problems that use the same base. Once I confirm that my method works, I use that method on all of the problems. Typical problems are: Base 2, Base e, Base 6, Base 10, Base 36, Base 64, converting between Base 2 and Base e, converting between Base 2 and Base 10, converting between Base 6 and Base 10, et cetera. May 1 '17 at 17:50