Answer
$$\frac{1}{2}(C-4A)$$
Work Step by Step
$\log_b2=A$ and $\log_b3=C$ $$X=\log_b\sqrt{\frac{3}{16}}$$ $$X=\log_b\Bigg(\frac{3}{16}\Bigg)^{1/2}$$
Apply the Power Rule here, we can move the exponent $1/2$ away for Quotient Rule usage later. $$X=\frac{1}{2}\log_b\frac{3}{16}$$
Now we can apply Quotient Rule for $\log_b\frac{3}{16}$ $$X=\frac{1}{2}(\log_b3-\log_b16)$$ $$X=\frac{1}{2}(\log_b3-\log_b2^4)$$
Again, use the Power Rule for $\log_b2^4$ $$X=\frac{1}{2}(\log_b3-4\log_b2)$$
Now we substitute A and C into X $$X=\frac{1}{2}(C-4A)$$