Question

• Solve \(x\) for \(x^{x\cdot x^x}=2\)
• Note that the Lambert W function \(W(x)\) is the inverse of \(f(x)=xe^x\).

Hint

It’s often useful solving a simpler problem first, such as \(x^x=2\).

Solution

$$x^{x\cdot x^x}=\left(x^x\right)^{\left(x^x\right)}$$

Let \(u=x^x\)

$$ \begin{align*} u^u&=2\\ u\ln u&=\ln 2\\ (\ln u)e^{\ln u}&=\ln 2 \end{align*} $$

We could let \(v=\ln u\), but it is clear that

$$ \begin{align*} \ln u&=W(\ln 2)\\ u&=e^{W(\ln 2)} \end{align*} $$

Or more generally, the solution to \(u^u=a\) is \(u=e^{W(\ln a)}\), because we will use it again.

$$ \begin{align*} x^x&=u\\ x&=e^{W(\ln u)}\\ &=e^{W\left(\ln e^{W(\ln 2)}\right)}\\ &=e^{W^2(\ln 2)} \end{align*} $$