Solving $|x-1|^{\log^2(x)-\log(x^2)}=|x-1|^3$

Question

Solve the equation:$$|x-1|^{\log^2(x)-\log(x^2)}=|x-1|^3.$$

There are three solutions of $x$: $10^{-1}$, $10^3$ and $2$. I obtained the first two solutions but I have been unsuccessful in getting $2$ as a solution. Please help.

Answer 1

The domain gives $x>0$ and $x\neq1$.

1. $|x-1|=1$.

With domain we get here $x=2$.

2. $\log^2x-\log{x^2}=3$.

Let $\log{x}=t$.

We need to solve now $t^2-2t=3$, which gives $t=-1$ or $t=3$, which is

$x=\frac{1}{10}$ or $x=1000$.

Finely we get the answer: $\{2,\frac{1}{10},1000\}$.