# “What if” math joke: the derivative of $\ln(x)^e$

Randall Munroe, the creator of xkcd in his latest book What if writes (p. 175) that the mathematical analog of the phrase "knock me over with a feather" is seeing the expression $\ln( x )^{e}$. And he writes regarding this expression: "it's not that, taken literally, it doesn't make sense - it's that you can't imagine a situation where this would apply."

In the footer (same page) he also states that "if you want to be mean to first year calculus students, you can ask them to take the derivative of $\ln( x )^{e}$. It looks like it should be "$1$" or something but it's not."

I don't get the joke. I think I am not understanding something correctly and I'm not appreciating the irony. Any help?

• It also appeared on the What If? website: what-if.xkcd.com/73 (see also footnote number 3). – Asaf Karagila Mar 7 '15 at 12:01
• How is this "it's not that, taken literally, it doesn't make sense -- it's that you can't imagine a situation where this would apply" not an explanation? Can you come up with a "natural" or "real world" mathematical situation where such an expression would appear? – Alexey Mar 7 '15 at 12:55
• @Alexey: Oh, now there's a challenge, if I ever saw one! But I'm sure MathSE is up to it. – Calum Gilhooley Mar 7 '15 at 14:08
• I don't like, or I'm still not used to this font - $e$ in exponent looks like $\varepsilon$. I noticed this only after wondering for 5-10 minutes about advantages of analysing usage of $\ln( x )^{\varepsilon}$ for $\varepsilon > 0$, until I noticed that it's not $\ln( x )^{\varepsilon}$, but $\ln( x )^{e}$. Still, I don't really get this joke as it looks for me as habit-based thing, like: people are used to eat things on plate with cutlery, but if "the thing" looks similar to the plate itself, it can be confusing or funny, but doesn't change the fact that it's ordinary food. – Kusavil Mar 7 '15 at 16:07

## 5 Answers

One is more accustomed to see something like $e^{\ln x}$, which is indeed equal to $x$. Its derivative is $1$.

In general, anytime you see exponentials elevated to a logarithm, you think this is going to simplify. In this case you have just a power of a logarithm, but that power is $e$, so it "looks" like an exponential, but of course it is not.

Not one of the best xkcd in my opinion though :P

Ah by the way, apparently there are a lot of people who are confused about xkcd jokes, and so explain xkcd was born… I used it a lot :D

• It would be meaner to ask the integral instead IMO. – AvZ Mar 7 '15 at 17:06
• This has nothing to do with xkcd, besides being by the same author. – ShreevatsaR Mar 7 '15 at 18:37
• @ShreevatsaR It also appeared on the xkcd website under a what-if segment (I thought it appeared on one of the stripes, though) – Ant Mar 7 '15 at 18:38
• @ShreevatsaR Sure it does. XKCD characters (with their same archetypal personalities) routinely pop up in What-If? illustrations. – bcrist Mar 7 '15 at 21:21
• @ShreevatsaR That's not exactly true either: xkcd.com/1416 But I understand and agree with your position; What If should not be referred to as xkcd or vice-versa. – bcrist Mar 8 '15 at 0:24

Good Lord of Purple Unicorns; I got the book a day ago, and I'm on page 172. XD

He means to say that many expressions like $e^{\ln(x)}$ and $\ln(e^x)$ equal $x$, but if you want to be mean to first year calculus students (owing to their naivety), they'll initially think it's a simple problem, but in fact the derivative of the expression $\ln( x )^{e}dx$ is

$$\frac{e(\ln(x))^{e-1}}{x}$$

• It's not even just naivete, I teach this level of students, and so many of them are just constantly desperate for a shortcut. It's actually rather distressing. – Ian Mar 7 '15 at 14:16
• ^Sorry I used the wrong word. I'm actually in position to make such comments, considering I'm only a tenth grader myself who as a real interest in Calculus :) @Ian – Kugelblitz Mar 7 '15 at 14:17
• Actually I would say that it is the naivety of the instructors that leads to students thinking that it is normal for problems have shortcuts and gentle answers. – DanielV Mar 7 '15 at 14:21
• @Kugelblitz - if you like pictures (and I do) and you are prepared to work through it Needham's Visual Complex Analysis has an explanation of contour integration that makes it seem utterly obvious. – Francis Davey Mar 7 '15 at 18:11
• ^Except in a case where there are many problems solvable by the one and the same shortcut; here taking more time to find a shortcut than solving a single question, may still be worth it. This is especially during specialised/topic-specific exams. – Kugelblitz Mar 8 '15 at 11:19

Well, since $e^{\ln(x)}$ and $\ln(e^x)$ equals $x$, and also since $\ln(x^a) = a \ln(x)$ your brain reasonably expects that there is some kind of simplification that applies to that expression, but there's not. It's the typical Randall joke.

The explanation is already there in what you quoted — Randall Munroe is simply and quite literally saying that an expression like $(\ln x)^e$ is extremely unlikely to crop up in any situation when working with mathematics either pure or applied.

It's not really a joke as such; he's just pointing out something that's absurdly improbable. (I think it's unfair to him to call it a joke and judge it as one.)

Apart from all that, there is also the matter of notation and familiarity. There are certain expressions that recur in mathematics, and outside of calculus textbook exercises it would be quite jarring to find oneself having to find the derivative of $c^x$ with respect to $c$, say, and it's almost psychologically harder to do than to find the derivative of $x^c$ with respect to $x$, even though they are mathematically entirely equivalent.

On notation, Halmos has alluded to this in his How to Write Mathematics:

As history progresses, more and more symbols get frozen. The standard examples are $e$, $i$, and $\pi$, and, of course, $0$, $1$, $2$, $3$, …. (Who would dare write “Let $6$ be a group.”?) A few other letters are almost frozen: many readers would feel offended if “$n$” were used for a complex number, “$\varepsilon$” for a positive integer, and “$z$” for a topological space. (A mathematician's nightmare is a sequence $n_\varepsilon$ that tends to zero as $\varepsilon$ becomes infinite.)

Related, from Milne's Tips for Authors (quoting Littlewood's Miscellany, p60):

It is said of Jordan's writings that if he had 4 things on the same footing (as $a,b,c,d$) they would appear as $a$, $M_3'$, $\varepsilon_₂$, $\Pi''_{₁,₂}$."

• Let $1$ be a group! (In all seriousness, I've seen and/or used $1$ for the trivial group, G's that look like $6$'s, and $e$ and $\pi$ appropriated for whatever was convenient at the moment (in one case forcing the user to write $\exp x$, and typically not even for a unit element or permutation).) – Vandermonde Mar 8 '15 at 15:25
• That reminds me... – J. M. isn't a mathematician Oct 9 '15 at 12:31

In former times we wrote $\sin x$, $\sin^2 x$, $\sin(2x)$. Then came Mathematica requiring us to write ${\tt Sin[x]}$, which is fine; but at the same time we are allowed to interpret ${\tt Sin[x]^2}$ as ${\tt (Sin[x])^2}$, which is not so obvious.

Therefore it is not at all clear what is meant by $\ln (x)^e$ outside of Mathematica. Is it $\log x^e$, $\bigl(\log x\bigr)^e$, or even something else? In any case: $\ \ln (x)^e$ is bad typography.