x^2 = 2^x. What is x ??

Yes, this is inspired by a homework question, the answer to which I obtained via graphing calculator. But it leads me to wonder how you would actually solve this equation. I started with taking logarithms of both sides but that only muddled things up more. One of the three answers is obvious (2; seeing as 2^2 = 2^2) and the others are x=4 and x= approx. -.767 ; how can you obtain those last two answers analytically?

The third value is -2*W(ln(2)/2)/ln(2), where W is Lambert’s W function (Google on that for some fun reading). I’ve seen an algebraic solution before, but it was very clever, and I don’t remember if it got the third value.

The “Lambert W function” is from Euler who named it after Lambert. It is specifically defined to be the solution to your equation. Other than that, there is no closed form solution (from this site ).
You will find that your problem is equivalent to solving the equation x[sup]x[/sup]=c.

If you find the answer unsatisfying, realise that, for instance, “sin” is also pretty much defined as the answer to an equation. It’s just that you’re used to those functions, so they feel like an answer, but the Lambert function doesn’t. Another answer is an explanation of why you can’t solve the equation in terms of familiar functions, but I left that in my other pants :slight_smile:

Here is a thread that I started about something similar to this:

In it, you’ll find lots of explanations and references (and yes, Lambert’s Function gets mentioned too). I also believe that there have been other similar threads before the one I started.

Well, there’s a way to get around that, actually. The sine function can be defined in terms of the exponential function on the complex plane, which itself is “really” a map from C to T[sub]1[/sub]C to C again.

T[sub]1[/sub]C is the tangent (complex) line to C, and since C is a topological vector space this tangent space can be canonically identified with C itself. This is the first map.

The second map is the diffeomorphism from a ball around the origin in the tangent space of a complex manifold at the identity to a ball around the identity in the manifold itself. Since C has no conjugate points, the domain is the entire tangent space. This is the second map.

Now that defines exp: C —> C, and as we all know

sin(z) = (exp(iz)-exp(-iz))/2i

Here’s a quick little calculator function–f(x)= - sqrt(2)^x

Start with an x, and keep plugging the answer back in the function. It’ll converge to the third value ( -.76666469…)