What is wrong with this attempt to solve $x^x = 2$?

Question

For $x^x= 2$, hence the solution is :

$x\cdot \ln(x) = \ln(2)$

$x\cdot\int(1/x)\,dx = \ln(2)$

$\int(x/x)\,dx = \ln(2)$

$x+c = \ln(2)$, where $c = 0$

so, $x = \ln (2)$

Can someone tell me why this solution is wrong?

Answer 1

The error was in the following step $$\int \frac{x}{x} dx$$ As you cannot transfer the $x$ into the integral that way because $x$ is not a constant. To bring it inside the integral, the new integral must evaluate to $x\ln x$. So the correct way to bring it inside will be

$$x\int\frac{1}{x}dx=\int(\ln x+1)dx$$

The actual solution requires the use of a special function called the lambert $W$ function where it has the property $$W(xe^x)=x$$ To begin with, you need to transform your equation into the form $ue^u$, which is easy enough $$x^x=2$$ $x=e^u$ $$e^{ue^u}=2$$ $$ue^u=\ln2$$ Using the lambert W function $$u=W(\ln2)$$ Substituting back in for $x$ $$\ln(x)=W(\ln2)$$ $$x=e^{W(\ln2)}$$ There are many solutions for the lambert W function, however the only one with a real answer is $W(\ln 2) \approx 0.444436091$ which makes the final answer $$x \approx 1.55961046946236$$

Answer 2

I believe that the error is that you incorrectly used the linearity of integration. You couldn’t move the x inside the integral of 1/x, because x isn’t a constant, it is the variable you are integrating with respect to.

Answer 3

It should start:

$x^x = 2$

$x\cdot \ln(x) = \ln(2)$

$x\cdot\int_1^x(1/t)\,dt = \ln(2)$

And then you're stuck! Sure you can put the $x$ inside the integral, but you're integrating FROM 1 to $x$ with respect to a placeholder variable (I've used $t$). And in fact, it's basically impossible to get unstuck without "cheating". And by cheating, I mean using the Lambert W function...

The Lambert W function often shows up in solving these types of problems. We can just define $W(x)$ to be the inverse of $xe^x$, and this turns out to be something for which numerical approximations are well-known. This video explores how to solve the problem using that at a relatively accessible level: https://www.youtube.com/watch?v=WWyMRmV1hLg .