Solve the equation $x(\log \log k - \log x) = \log k$ I want to solve this equation by expressing $x$ in function of $k$. Is it possible?

Thanks.
Yes, but you're gonna need the Lambert W function.

\begin{align}
x(\log \log k - \log x) &amp;= \log k\\
\log x - \log \log k &amp;= \frac{-\log k}x\\
\frac x{\log k} &amp;= e^{-(\log k)/x}\\
-1 &amp;= \frac{-\log k}{x}\cdot e^{-(\log k)/x}\\
W(-1) &amp;= \frac{-\log k}{x}\\
x &amp;= \frac{-\log k}{W(-1)}\\
\end{align}

Note that $W(-1)$ is a complex number.
differentiating both sides with respect to x and taking  $$loglogk = p$$
$$d(px)/dx - d(xlogx)/dx = 0$$
$$p - x*d(logx)/dx - logx = 0$$
$$p - 1 - logx = 0$$
$$logx = p - 1$$
$$x = e^p/e$$
$$x = (logk)/e$$