## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

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