Answer
$2-\displaystyle \frac{1}{2}\log_{6}(x+1)$
Work Step by Step
$\displaystyle \log_{6}\left(\frac{36}{\sqrt{x+1}}\right)=$
...apply the Quotient Rule: $\displaystyle \quad \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$
=$\log_{6}36-\log_{6}\sqrt{x+1}$
... write $36$ as $6^{2},\quad\sqrt{x+1}=(x+1)^{1/2}$
$=\log_{6}6^{2}-\log_{6}(x+1)^{1/2}\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$=2\displaystyle \log_{6}6-\frac{1}{2}\log_{6}(x+1)\quad $...apply the basic property: $\log_{b}b=1$
$=2-\displaystyle \frac{1}{2}\log_{6}(x+1)$
