Answer
$ \displaystyle \frac{1}{2}\log_{a}(c-d)-\frac{1}{2}\log_{a}(c+d)$
Work Step by Step
$\displaystyle \log_{a}\frac{c-d}{\sqrt{c^{2}-d^{2}}}= \qquad$ ... apply $ \displaystyle \log_{a}\frac{M}{N}=\log_{a}M-\log_{a}N$
$=\log_{a}(c-d)-\log_{a}\sqrt{c^{2}-d^{2}}$
.... rewrite $\sqrt{c^{2}-d^{2}}$ as $[(c-d)(c+d)]^{1/2}$
$=\log_{a}(c-d)-\log_{a}[(c-d)(c+d)]^{1/2}$
... apply $\log_{a}M^{p}=p\cdot\log_{a}M$
$=\displaystyle \log_{a}(c-d)-\frac{1}{2}\log_{a}[(c-d)(c+d)]$
... apply $\log_{a}(MN)=\log_{a}M+\log_{a}N$
$=\displaystyle \log_{a}(c-d)-\frac{1}{2}\log_{a}(c-d)-\frac{1}{2}\log_{a}(c+d)$
$= \displaystyle \frac{1}{2}\log_{a}(c-d)-\frac{1}{2}\log_{a}(c+d)$
