How to solve "impossible" equations using Lambert's W function

I came across this question on Twitter. Questions of this sort are fairly common, but the method of solution is not very well known, so I thought that it was worth a few words of explanation.

The easy answer is that the equation cannot be solved using standard algebraic functions; one must use numerical methods such as Newton's method to approximate the solutions. But it can be solved algebraically if we add some more functions to our repertoire, just as the introduction of the square root function allowed us to solve quadratic equations.

In this case, the "magic" function is called the Lambert W function. It is defined as the inverse of the function f(x) = x * e^x. In order to use this function, we must manipulate the equation into the form x * e^x = y, then we invert to obtain x = W(y).

Let's use this idea to solve x^3/24 - ln(x) = 0. We get rid of the logarithm by substituting x = e^t, yielding the equation e^(3t)/24 - t = 0. This can be rewritten as t * e^(-3t) = 1/24, or -3t * e^(-3t) = -1/8.

Now here is where the magic happens. We use Lambert's W function to invert this equation, and we get -3t = W(-1/8), or t = -W(-1/8)/3. Since x = e^t, our final answer is x = e^(-W(-1/8)/3).

An astute reader might graph the equations y = x^3 / 24 and y = ln(x), and observe that the curves cross in two points. So why did I only give one solution? Well, it turns out that the function f(x) = x * e^x is not one-to-one; it's increasing when x > -1 and decreasing when x is less than -1. So the Lambert W "function" actually has two branches, and the two branches yield the two solutions to the equation. This is similar to the fact that every positive number has two square roots.