Solve for x : $(13)^{\sqrt{x}} = 4^4 - 3^4 - 6.$

Given,

$\Rightarrow (13)^{\sqrt{x}} = 4^4 - 3^4 - 6 \\[1em] \Rightarrow (13)^{\sqrt{x}} = 256 - 81 - 6 \\[1em] \Rightarrow (13)^{\sqrt{x}} = 169 \\[1em] \Rightarrow (13)^{\sqrt{x}} = 13^2 \\[1em] \Rightarrow \sqrt{x} = 2$

Squaring both sides of the above equation, we get :

$\Rightarrow (\sqrt{x})^2 = 2^2 \\[1em] \Rightarrow x = 4.$

Hence, x = 4.