Answer
$\frac{1}{2}C - 2A$
Work Step by Step
Note that $16=2^4$.
Thus, the given expression is equivalent to:
$=\log_b{\sqrt{\left(\frac{3}{2^4}\right)}}$
Use the rule $\sqrt{m} = m^{\frac{1}{2}}$ to obtain:
$= \log_b{\left[\left(\frac{3}{2^4}\right)^{\frac{1}{2}}\right]}$
RECALL:
(1) $\log_b{(\frac{m}{n})} = \log_b{m} - \log_b{n}$
(2) $\log_b{mn} = \log_b{m} + \log_b{n}$
(3) $\log_b{(m^n)} = n \cdot \log_b{m}$
Use rule (3) above to obtain:
$=\frac{1}{2} \cdot \log_b{\left(\frac{3}{2^4}\right)}$
Use rule (1) to obtain:
$=\frac{1}{2}\left(\log_b{3} - \log_b{(2^4)}\right)$
Use rule (3) to obtain:
$=\frac{1}{2}\left(\log_b{3} - 4\log_b{2}\right)$
With $\log_b{2} = A$ and $\log_b{3}=C$, the expression above is equivalent to:
$=\frac{1}{2}(C-4A)
\\=\frac{1}{2}C - 2A$
