Answer
$2\log{x}+\frac{1}{2}\log{(x^3+1)}$
Work Step by Step
Recall:
(1) $\sqrt[m]{a}=a^{\frac{1}{m}}$
(2) $\log_a {x^n}=n\cdot \log_a {x}$.
(3) $\log_a{xy}=\log_a{x} +\log_a{y}$
(4) $\log_a{\frac{x}{y}}=\log_a{x} -\log_a{y}$
($\log_a{M}=\log_a{N} \longrightarrow M=N$.)
Using Rule(3): $\log{x^2\sqrt{x^3+1}}=\log{x^2}+\log{\sqrt{x^3+1}}.$
Using Rule(2): $\log{x^2}+\log{\sqrt{x^3+1}}=2\log{x}+\frac{1}{2}\log{(x^3+1)}$
