$x\ln (\frac{2}{x})=k(a-x) \ln (\frac{2}{a-x})$

where $a$ and $k$ are positive constants. $a$ is usually small, say, $0< a<0.1$ and $x\in (0,a)$.

There are ways to calculate numerical solutions. However, as it is a intermediate step of an optimization problem, I do need closed form results to move forward with some strict proof. Does that exist?

One observation is that function $f(x)=x\ln (\frac{2}{x})$ is increasing when $x\in (0,2/e)$, thus with $0< a<0.1$ and $x\in (0,a)$, $f(x)$ is increasing. I don't know if this helps.