Solve for $x$ \[ 27^{-x+2}=81 \]

Step 1 ：Solve for $x$ in the equation $27^{-x+2}=81$.

Step 2 ：Use the property of exponents that says $(a^b)^c = a^{bc}$ to simplify the left side of the equation to $3^{3(-x+2)}$.

Step 3 ：Set the simplified equation equal to $81$, which is $3^4$, to get $3^{3(-x+2)} = 3^4$.

Step 4 ：Solve the equation for $x$ to get the solutions $x = \frac{2}{3}$, $x = \frac{2}{3} - \frac{2i\pi}{3\log(3)}$, and $x = \frac{2}{3} + \frac{2i\pi}{3\log(3)}$.

Step 5 ：However, in the context of this problem, we are likely only interested in the real solution.

Step 6 ：The solution to the equation is $\boxed{\frac{2}{3}}$.