How do we know x has to be a (3^n)? Is it because the LCM of 3 and 81 is 3? How do we decide what to try to brute force our way into this?
The whole point is to express 3^{81} in a form where both the base and the exponent matches. You can keep distributing the exponent to the base till they match.
3^{81} = (3^9)^9 = (3^3)^{27} = 27^{27}