Answer
$2\log{x}+\frac{1}{2}\log{(x^3+1)}$
Work Step by Step
The given expression is equivalent to:
\begin{align*}
&=\log{\left(x^2(x^3+1)^{\frac{1}{2}}\right)}\\
\end{align*}
Recall:
(1) $\log_a{(mn)}=\log_a{m} + \log_a{n}$
(2) $\log_a{\left(\frac{m}{n}\right)}=\log_a{m} - \log_a{n}$
(3) $\log_a{a^m}=m\log_a{m}$
Use rule (1) above to obtain:
\begin{align*}
\log{\left(x^2(x^3+1)^{\frac{1}{2}}\right)}&=\log{\left(x^2\right)}+\log{(x^3+1)^{\frac{1}{2}}}
\end{align*}
Use rule (3) above to obtain:
\begin{align*}
\log{(x^2)}+\log{(x^3+1)^{\frac{1}{2}}}&=2\log{x}+\frac{1}{2}\log{(x^3+1)}
\end{align*}
