2010 AIME II Problem 5

Positive numbers x, y, and z satisfy x y z=10^{81} and \left(\log _{10} x\right)\left(\log _{10} y z\right)+ \left(\log _{10} y\right)\left(\log _{10} z\right)=468.
Find \sqrt{\left(\log _{10} x\right)^{2}+\left(\log _{10} y\right)^{2}+\left(\log _{10} z\right)^{2}}.