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

Question

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?

Answer 1

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

Answer 2

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.

Answer 3

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

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

Answer 4

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 $a$, $M_3'$, $\varepsilon_₂$, $\Pi''_{₁,₂}$."