$\mathbf{Solution}$: Use the property of the log function to get that $LHS = log_2(a^{x_1+x_2+x_3} + a^{x_1+x_2} + a^{x_3+x_2}+a^{x_1+x_3} + a^{x_1} + a^{x_2}+a^{x_3} + 1)$. Use that $x_1+x_2+x_3 = 0$, this simplifies to $LHS = 2 + (a_{x_1} +\frac{1}{a^{x_1}}) + (a_{x_2}+ \frac{1}{a^{x_2}})+(a_{x_3}+ \frac{1}{a^{x_3}}) $. Using AM-GM we get that this expression is greater than or equal to $2+2+2+2$ =$8$. So the $LHS \geq 8$. Take log on both sides to get that $log_2(LHS) \geq log_2(8) =3$. $\blacksquare$