Question

Find all the solutions of the equation $|x-1{|}^{(\log x{)}^2-\log x^2}=|x-1{|}^3$, where base of logarithm is $10$.

Answer 1

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

$x>0,$

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

$(\log x-3)(\log x+1)=0$

$\log x=3or-1$

$x={10}^{3}or{10}^{-1}$

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