Power tower question $$x^{x^{x^{.^{.^{.}}}}} = 8$$
Then how to solve for x?
I first tried like this
$x^8=8$ but I don't get any way to solve.

There is no way to obtain an analytical solution in terms of elementary functions.
However, one can find an expression in terms of the Lambert W function. This expression evaluates to $8=-\frac{W(-ln(x))}{ln(x)}$. This expression can be solved using numerical methods.
However, as noted by others you may notice that infinite tetration (technical word for power-tower) of $x$ converges if and only if $x \in [e^{-e},e^{1/e}]$. Therefore, your may use your positive real solution to $x$ for $x^8=8$. This will be identical to the solution to your question since:
$$e^{-e}<x<e^{1/e}$$
Therefore, $x$ cannot converge to any other value.
