Question

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

Answer 1

if the above equatio is right then their powers must be equal,so

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

$(logx{)}^2-2logx-3=0$

let $logx=t$

$t^2-2t-3=0$

$(t-3)(t+1)=0$

$t=3,-1$

$logx=3,logx=-1

$x={10}^3,\frac{1}{10}$