jmc

Rewriting both sides with $3$ as the base, we have $81^{2x}=(3^4{)}^{2x}=3^{8x}$ and $27^{3x-4}=(3^3{)}^{3x-4}=3^{9x-12}$, and so our equation is $$3^{8x}=3^{9x-12}.$$Then, setting the exponents equal to each other, we obtain $$8x=9x-12.$$This yields our solution $\boxed{x=12}.$