# What's the deal with Lambert W function questions?

I have seen a large amount of questions that are:

solve $x^2=2^x$

Why are there so many of these questions? Who's assigning these questions? Why did I never see them when I took high school algebra? Are they just for personal interest? If so, why are so many of them so convoluted looking?

I have never seen the Lambert W function in my experiences in undergrad/now grad school. I see it fairly routinely on MSE. Why is the Lambert W function so overrepresented on MSE?

• It's not overrepresented, here. It's not even underrepresented elsewhere, simply folks there wouldn't understand it. – Professor Vector Nov 20 '17 at 19:43
• I share this sentiment. Lambert W was totally absent during my undergrad education as well as grad school (the latter is hardly a surprise). The non-elementary functions we learned about were the likes of Bessel functions, Gamma, Erf and such that were actually needed in various places (physics, number theory, whatnot). I guess times have changed??? – Jyrki Lahtonen Nov 20 '17 at 19:45
• I can't say it was totally missing from my undergraduate studies. But I think that it was only brought up once in either measure theory or functional analysis. – Asaf Karagila Nov 20 '17 at 20:37
• I would not imagine anyone is assigning such things. We get an enormous slopover from other websites, several kids see something on a website and post it here, paraphrasing in random ways. – Will Jagy Nov 20 '17 at 20:43
• My guess is that some people wrongly copy the problem by either omitting details like $x$ is a natural number or they are asked how much solutions are there instead of actually needing to solve the equation. – kingW3 Nov 20 '17 at 20:47
• What @kingW3 said; or, they didn't notice that they were being asked to solve numerically. Some of it may just be curiosity. You sketch the graph of $y=e^x$, you sketch the graph of $y=-x$, you see they intersect somewhere, you wonder just where that somewhere is. Or, you try to find a local maximum for some innocent-looking function by setting the derivative to zero, and you get an equation custom-made for Lambert W. – Gerry Myerson Nov 20 '17 at 22:51
• I don't find the Lambert function very interesting as opposed to other non-elementary functions like Gamma, Zeta, polylog, elliptic/theta/hypergeometric functions. When I see an equation like the one in your question, I don't think beyond techniques for numerical solution (like @GerryMyerson said above). I wonder if Lambert function has any interesting connections with other more popular functions. Questions involving these equations do not appear to be well thought out. – Paramanand Singh Nov 21 '17 at 5:55
• The first time I saw Lambert W was on Math.SE. Since then it's just popped up all over the place. – Antonio Vargas Nov 21 '17 at 6:01
• I'm a combinatorist and although I know that some combinatorists use it for some stuff about tree counting, I think it came up once or twice across all undergraduate or graduate courses I've taken. It has come up a bunch in musings I've had as a high schooler, but nothing for coursework or research. I graduated in 2016, so I don't think this is a time thing. – Stella Biderman Nov 21 '17 at 17:04
• I think certain topics attract more attention in online discussion groups than in formal classes, and this has been true since at least the early 1990s (on sci.math). Other examples are tetration, proofs that require the Axiom of Choice, non-computable functions such as the Busy Beaver function, fractals (self-similar kinds), deconstructing (and often just "disproving") the Cantor diagonal argument for uncountability of the reals, etc. If anything, in the last 7 or 8 years with math stack exchange and mathoverflow, online math discussions have gotten MUCH closer to standard textbook topics. – Dave L. Renfro Nov 21 '17 at 19:36
• Now is about the time when Newton's Method is being taught in Calc 1. Possibly they are popping up in that context, or in the context of the MVT being used to demonstrate the existence of unique solutions. – Scott H. Nov 22 '17 at 18:49
• The natural symmetry of the variable and constants in $x^2 = 2^x$ seems a likely cause for curiosity. As I recall when I first had access to a computer algebra system it was among the first non-trivial equations I asked it to solve, expecting an answer in terms of known functions, only to discover the lambert W function. This might play a part in its rise in popularity post-internet. – CyclotomicField Nov 25 '17 at 13:10
• @Cyclo, I believe there is an answer in terms of known functions, viz., $x=2$. – Gerry Myerson Nov 26 '17 at 0:34
• To second the comment about combinatorics, W shows up when studying Bell and Stirling numbers and (set) partitions; for instance, it turns out to be relevant to calculating (approximate) means and modes of certain distributions associated with them. See volume 4 of The Art of Computer Programming for some details. – Steven Stadnicki Nov 29 '17 at 19:54
• The published version of @DuchampGérardH.E.'s article: Corless et al. - On the Lambert $W$-function. (Note that 'et al.' hides Knuth.) – LSpice Nov 29 '17 at 21:01

I don't know, it seems to me a very natural question to arise, not from homework, but from a student's own curiosity, while first learning algebra. Maybe they tried plotting the functions on a graphing calculator and see that there is an answer, but are stuck trying to find it themselves with the tools they know. Maybe they asked their teacher and were told, "it's impossible," giving the problem an air of mystery that also drives interest in dividing by zero, formulas for roots of high-degree polynomials, summation methods for divergent series, etc.

The other answers show, that there is a certain universality to human curiosity (especially the instinct to generalize). When you know just enough high school math to start playing around with it, there seem to be a very limited number of questions that people ask themselves. When you start learning about powers, especially when $x$ starts going into the exponent, you go and investigate the difference between $a^x$ and $x^a$ (or even make a few mistakes this way).

I myself noticed a classmate making mistake between $x^2$ and $2^x$ and immediately asked myself when he'd actually be right. I noticed there is a third root in addition to $2$ and $4$ (got there by plotting), and was fascinated to find out it's "impossible" (along the way, discovered the power-tower solution and found the intersection manually with bisection on graph paper). Generalization to $a^x=x^a$ was even more interesting (discovering that $a=e$ special case has a double solution in positives - the magic $e$ strikes again). I even went so far to try to solve this in excel (high-school, didn't know the tools), and made a t-shirt with the negative solution $-0.76666469596212309311...$ printed on it.

Don't forget teenagers tend to think big and exaggerate. Dealing with $a^x$ and $x^a$ (and differences in how you differentiate them), you try being clever by going "what about $x^x$? Which rule applies now?". This of course again quickly leads to Lambert's function.

I'd say, these are the "popular" interesting math questions high-schoolers reinvent over and over again independently:

• Reciting $\pi$ (for some reason, always $\pi$, almost never $e$ or some other constant - there's just something attractive about it).
• $x^x$, $x=e^x$ and variations (Lambert function type)
• $x=\cos x$, $ax=\sin x$ and $x=\tan x$ (trig-type transcendentals, shocked to find out it can't be analytically solved)
• Generalized mean $((x^n+y^n)/2)^{1/n}$, maybe leading to the limiting case, which gives you the geometric mean. Alternatively, generalized Pythagorean theorem (no no division involved).
• Infinite sums (usually related to Riemann Zeta, geometric or similar "obvious" sequences, sometimes leading to $1+2+3+\cdots$), again, shocked to see it's not straight forward to compute.

When I was young, I amused myself with various mathematical problems I came up with on my own, including finding numerical solutions to interesting equations such as $x^x = 2$. At that time, I would just perform binary search to find the solution to the precision on my calculator. It was of interest to me to find out that the solution could be expressed in terms of the Lambert W function, despite not appearing to be in the same form.

As for your specific equation in your question, if $x$ is a positive real, it can be proven elementarily that $x^2 = 2^x$ iff $x = 2$ or $x = 4$. In particular, all you need is to show that the function $f : \mathbb{R}^+ \to \mathbb{R}$ defined as $f(x) = x^{1/x}$ for real $x > 0$ has a unique maximum at $e$. So sometimes the question may have a simple solution that has nothing to do with the Lambert W function.

• There's also a negative solution $\approx-0.7666647$, so your "iff" is wrong. – Ruslan Dec 4 '17 at 10:09
• @Ruslan: Argh; you are right. I was thinking "positive real" but wrote only "real". I've edited my post, thanks! – user21820 Dec 4 '17 at 10:40

I think these questions became popular with the rise of the calculator. When we first started using Maple and Mathematica in the classroom, we needed problems that still required thinking. (I was against the tendency to create complicated problems just so Maple wouldn't instantly solve them and force the student to think the problem through. It seemed an easier, much cheaper solution to just remove the CAS.) One way was to use unfamiliar functions. This became a never-ending cycle, because the instant we found something Maple didn't know, the people at Waterloo would add it.

If you want a student to understand how graphs and solutions are related, then these are pretty good problems for that. You can test understanding of the inverse function theorem and some Taylor series things, too, with Lambert-type problems.