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?
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:
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.
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.
The most common way to solve equations of one variable in closed form is to guess particular solutions or to apply the inverse operations that can be read from the equation.
The latter method is applicable if the function of the equation is a composition of functions of one variable with known inverse functions. Algebraic operations are algebraic functions.
The elementary functions are generated from their arguments by applying finite numbers of $\exp$, $\ln$ and/or unary or multiary $\mathbb{C}$-algebraic functions.
If we have an equation of the form $$A(f_1(x),...,f_n(x))=y,\tag{1}$$ where $n>1$, $A$ an algebraic function of $n$ variables, $y$ a constant or a variable, and $f_1,...,f_n$ are elementary functions of one variable that are algebraically independent, we don't know how to solve the equation by rearranging for $x$ by applying only elementary functions we can read from the equation.
It can be proven that the function on the left-hand side of the equation $$A(x,e^x)=y\tag{2}$$ cannot have partial inverses that are elementary functions.
Moreover, for functions $A$ for which equation (2) leads to an equation $P(x,e^x)=0\ $ (3), where $P$ is an irreducible polynomial function of two complex variables with algebraic coefficients, it was proven that equation (3) doesn't have nonzero solutions $x$ that are elementary numbers.
That means, equations of the form (2) cannot be solved by applying elementary functions, except by $x=0$.
But such kinds of equations occur more frequently. Lambert W ist the most simple, most described and most known representative of inverse relations for equations of the type of equation (1).
[Mező/Keady 2015]:
"The survey paper of Corless et al. [6] describes a large number of applications of W. This function appears in the combinatorial enumeration of trees, in the jet fuel problem, in enzyme kinetics, or in the solution of delay differential equations, just to mention a few areas. Some specific applications of W in the investigation of solar wind comes from Cranmer [9], some applications in electromagnetic behavior of materials are given by Houari [13], other appearances in electromagnetics are investigated by Jenn [17]. It is known that Wien’s displacement constant can be expressed by W, and in the discussion of capacitor fields W also appears [34].
The Lambert function has applications in quantum statistics, too [33]. A simple mathematical application connects W to the distribution of primes via the Prime Number Theorem [36]."
[Mező/Baricz 2017]:
"In 2002 R. M. Corless and D. J. Jeffrey wrote that the Lambert W function is the simplest example of the root of an exponential polynomial; and exponential polynomials are the next simplest class of functions after polynomials". Thus, says B. Hayes [12], W is in some sense the smallest step beyond the present set of elementary functions". Based on these thoughts, it can happen that the complexity" of the root of an exponential polynomial strongly depends on the degree of the polynomial, as is the case for the classical polynomials. Speculatively, a new class of functions would arise in which W is just the simplest" member (in the Corless-Jeffrey sense) ..."
[Vazquez-Leal et al. 2020]:
"Later on, J.H. Lambert, in 1758, provided the principles for a new transcendental function (𝑊(𝑥)) that was almost unknown until 1990s, including its applications to science and engineering. This function was fully developed until 1925 by Pólya and Szegö [51] moreover, thanks to the algorithms implemented by Corless & developers [57], this function began to be widely employed by the end of 20th century."
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How can we show that $A(z,e^z)$ and $A(\ln (z),z)$ have no elementary inverse?
[Mező/Keady 2015] Some physical applications of generalized Lambert functions. 2015
