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