How to start solving this Logarithmic problem?

Question

If $abc= 2^6$, $a, b, c \ge 0$, $\log_2 (a)\log_2 (bc)+\log_2 (b)\log_2 (c)= 10$, find $\sqrt{((\log_2 (a))^2 + (\log_2 (b))^2 + (\log_2 (c))^2}$

Answer 1

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 $$\sqrt{A^2+B^2+C^2}=\sqrt{16}=4$$