Question

Find the value of $x$ if ${|x-1|}^{({log}_{10}x{)}^2-lo{g}_{10}{x}^2}={|x-1|}^3$

Answer 1

${|x-1|}^{{\left({log}_{10}x\right)}^2-{log}_{10}{x}^2}={|x-1|}^3$

As the bases are same, the power must be equal.

$\Rightarrow {\left({log}_{10}x\right)}^2-{log}_{10}{x}^2=3$

$\Rightarrow {\left({log}_{10}x\right)}^2-2{log}_{10}x=3$

Let ${log}_{10}x=k$

$k^2-2k-3=0$

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

$\Rightarrow k=3,k=-1$

$\therefore$ ${log}_{10}x=3,-1$

if ${log}_{10}x=3$

$\Rightarrow x=1000$

if ${log}_{10}x=-1$

$\Rightarrow x=\frac{1}{10}=0.1$