Answer
$4\displaystyle \ln x+\frac{1}{2}\ln(x^{2}+3)-5\ln(x+3)$
Work Step by Step
$\displaystyle \ln[\frac{x^{4}\sqrt{x^{2}+3}}{(x+3)^{5}}]=\quad $...apply the Quotient Rule: $\displaystyle \quad \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$
$=\ln(x^{4}\sqrt{x^{2}+3})-\ln(x+3)^{5}$
$\quad $...apply the Product Rule: $\quad \log_{b}(MN)=\log_{b}\mathrm{M}+\log_{b}\mathrm{N}$
$=\ln x^{4}+\ln\sqrt{x^{2}+3}-\ln(x+3)^{5}\quad $...write the root in exp. form
$=\ln x^{4}+\ln(x^{2}+3)^{1/2}-\ln(x+3)^{5}$
$\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$=4\displaystyle \ln x+\frac{1}{2}\ln(x^{2}+3)-5\ln(x+3)$
