Answer
$\{1+\sqrt[3]{2}\}$ or $\{2.260\}$
Work Step by Step
(*) For the equation to be defined, it must be true that $x \gt 1$
...Apply$: \quad \log_{a}M^{r}=r\log_{a}M\quad $
$\log(x-1)=\log 2^{1/3}$
...Apply$: \quad $If $\log_{a}M=\log_{a}N,$ then $M=N$
$x-1=2^{1/3}$
$x=1+\sqrt[3]{2}\approx 2.25992$
The solution satisfies (*), so the solution set is
$\{1+\sqrt[3]{2}\}$ or $\{2.260\}$
