How to start solving this Logarithmic problem? If $abc= 2^6$, $a, b, c \ge 0$, $\log_2 (a)\log_2 (bc)+\log_2 (b)\log_2 (c)= 10$, find $\sqrt{((\log_2 (a))^2 + (\log_2 (b))^2 + (\log_2 (c))^2}$

Using the substitution suggested in the comments gives the equations
\begin{cases}
A+B+C=6\\
AB+AC+BC=10
\end{cases}
We can then square the first equation to give
\begin{align}
(A+B+C)^2
&=A^2+B^2+C^2+2(AB+AC+BC)\\
&=A^2+B^2+C^2+2\cdot10\\
&=A^2+B^2+C^2+20\\
&=6^2\\
&=36\\
\end{align}
Thus we get
$$A^2+B^2+C^2=36-20=16$$
Note that you want the quantity
$$\sqrt{A^2+B^2+C^2}=\sqrt{16}=4$$
