If $(9^{4})^{x} \equiv 12 \pmod{23}$, then how do I find $x$?
I have tried solving this, but I was thinking if there is a step by step formula. I know that any number from the group order may suffice.
Will I still be able to find $x$ when the numbers involved are huge, such as, a large prime instead of $23$ and a large number instead of $4$?
$$3^{8x-1}\equiv4\pmod{23}\equiv3^3$$
$$\iff3^{8x-4}\equiv1$$
$3^2\not\equiv1,3^3\equiv4,3^{11}\equiv9(4^3)\equiv1$
$$\implies8x-4\equiv0\pmod{11}$$