Using the following properties:
[tex]\begin{gathered} \log _z(x)^y=y\log _z(x) \\ \log _z(\frac{x}{y})=\log _z(x)-\log _z(y) \\ \sqrt[z]{x^y}=x^{\frac{y}{z}} \\ \log _z(x\cdot y)=\log _z(x)+\log _z(y) \end{gathered}[/tex]
so:
[tex]\begin{gathered} \log _{12}(\sqrt[3]{\frac{12+x}{144x}})=\log _{12}(\frac{12+x}{144x})^{\frac{1}{3}}=\frac{1}{3}\log _{12}(\frac{12+x}{144x}) \\ \\ so\colon \\ \frac{1}{3}\log _{12}(\frac{12+x}{144x})=\frac{1}{3}\log _{12}(12+x)-\frac{1}{3}\log (144x) \\ \\ \frac{1}{3}\log _{12}(12+x)-\frac{1}{3}\log (144x)=\frac{1}{3}\log _{12}(12+x)+\frac{1}{3}\log (144)-\frac{1}{3}\log _{12}(x) \end{gathered}[/tex]
Therefore, the answer is:
[tex]\log _{12}(\sqrt[3]{\frac{12+x}{144x}})=\frac{1}{3}\log _{12}(12+x)+\frac{2}{3}-\frac{1}{3}\log _{12}(x)[/tex]
