How do I cube/square a logarithm?

1) Remember that $\,\log x^n=n\log x\,$

2) Now just note that $\,\log \sqrt[3] x=\log x^{1/3}\,$

Anyways, it is not true in general that $\,\log \sqrt[3] x=\sqrt[3]{\log x}\,$

You can write $\log \sqrt[3] x$ in terms of $\log x$ using the laws of logarithms. So substitute $u=\log x$ and solve for $u$ (after rearranging you have a cubic equation in $u$) then substitute back and solve for $x$.

---

Edit: Your error in your working is thinking that $a^3 = \log x^3$. In fact, $a^3 = (\log x)^3$.