#### Answer

$2 \log (x)+\dfrac{1}{2}\log(x^3+1)$

#### Work Step by Step

The given expression can be re-arranged as:
$ \log(x^2 \sqrt {x^3+1}=\log[x^2 (x^3+1)^{1/2}]$
Use $\log_a{(mn)}=\log_a{m} + \log_a{n}$ to obtain:
$\log[x^2 (x^3+1)^{1/2}]=\log (x^2) +\log (x^3+1)^{1/2}$
Use $\log_a{a^m}=m\log_a{m}$ to obtain:
$ \log(x^2) +\log (x^3+1)^{1/2}=2 \log (x)+\dfrac{1}{2}\log(x^3+1)$