Equation:
$$a = {x^{x^{x^{.^{.^{.}}}}}}$$

Take $ln$ of both sides, then use the power rule in right and plug $"a"$ in ${x^{x^{x^{.^{.^{.}}}}}}$:

$$ln(a) = ln({x^{x^{x^{.^{.^{.}}}}}})$$
$$ln(a) = {x^{x^{x^{.^{.^{.}}}}}} \cdot ln(x)$$
$$ln(a) = a \cdot ln(x)$$

Divide both sides by "$a"$, then rearrange left side for using power rule afterwards:
$$\frac{ln(a)}{a} = ln(x)$$
$$\frac{1}{a}ln(a) = ln(x)$$

Use power rule in left, then take $exp$ of both sides and lastly, rewrite left side as a root:
$$ln(a^\frac{1}{a}) = ln(x)$$
$$a^\frac{1}{a} = x$$
$$\sqrt [a]{a} = x$$