### Instruction: Solve for $x$: $$81^{2x} = 27^{3x - 4}.$$.### Response: I notice that both 81 and 27 are powers of 3, so I can rewrite them using exponent rules: $$81^{2x} = 27^{3x - 4} \implies (3^4)^{2x} = (3^3)^{3x - 4}.$$
Now I can simplify the exponents by multiplying them: $$(3^4)^{2x} = (3^3)^{3x - 4} \implies 3^{8x} = 3^{9x - 12}.$$
Since the bases are the same, I can set the exponents equal to each other and solve for $x$: $$8x = 9x - 12 \implies x = 12.$$