Proving (2nd order) convergence of fixed point iteration

Question

I want to find the (unique) $x\in\mathbb R$ which solves the equation $x\exp(x)=a$, with $a>0$ given. After experimenting a bit with Excel, I think that the fixed point iteration $$x_{k+1}=\ln\left(\frac a{x_k}\right)$$ converges quite fast. Specifically, I want to prove the convergence is of second order locally, which means there is a constant $c>0$ such that for all $x_0$ close enough to the actual solution $x_*$ and for all $k\in\mathbb N$ the inequality $$\left|\ln\left(\frac a{x_k}\right)-x_*\right|\leq c\cdot\left|x_k-x_*\right|^2$$ holds. How could I prove this? I neither know how to use the "$x_0$ close enough" part nor do I have a closed expression for $x_*$ which I could just plug in and see what happens.