Answer
$\displaystyle \frac{1}{2}A-\frac{3}{2}C $
Work Step by Step
$\displaystyle \sqrt{\frac{2}{27}}=(\frac{2}{27})^{1/2}=(\frac{2}{3^{3}})^{1/2}=\frac{2^{1/2}}{3^{3/2}}$
$\displaystyle \log_{b}\sqrt{\frac{2}{27}}=\log_{b}\frac{2^{1/2}}{3^{3/2}}$
...apply the Quotient Rule: $\displaystyle \quad \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$
$=\log_{b}2^{1/2}-\log_{b}3^{3/2}$
$\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$=\displaystyle \frac{1}{2}\log_{b}2-\frac{3}{2}\log_{b}3$
... use the definitions of A and C
$=\displaystyle \frac{1}{2}A-\frac{3}{2}C $
