$\log_j(\log_j(\log_j(x)))=\log(x);\ \ j=?$ $\log_j(\log_j(x))=\log(x)$ has solution $j=x^{\exp-W(\log^2(x))}$ for real $x\neq0$, where $W=$ Lambert W function.
But what is the solution to $\log_j(\log_j(\log_j(x)))=\log(x)$? Mathematica can't do it - can it be done?

I presume that the only possible solutions corresponds to $j=\frac 1k$ to make the equation
$$\frac{\log \left(-\frac{\log \left(-\frac{\log (x)}{\log (k)}\right)}{\log
   (k)}\right)}{\log (k)}+\log (x)=0$$ Numerically solving for $x$ for a given $k$, the following (not very exciting) results
$$\left(
\begin{array}{cc}
k & x_k \\
 2 & 0.743810 \\
 3 & 0.773972 \\
 4 & 0.806977 \\
 5 & 0.831510 \\
 6 & 0.849777 \\
 7 & 0.863823 \\
 8 & 0.874964 \\
 9 & 0.884035 \\
 10 & 0.891583 \\
 20 & 0.930106 \\
 30 & 0.945785 \\
 40 & 0.954705 \\
 50 & 0.960604 \\
 60 & 0.964859 \\
 70 & 0.968106 \\
 80 & 0.970684 \\
 90 & 0.972792 \\
 100 & 0.974555 \\
1000 & 0.994683
\end{array}
\right)$$
