Question

Solve the equation
${|x-1|}^{{\log }_e^{2}x-{\log }_e{x}^2}={|x-1|}^3$.
Find the number of solutions greater than zero .

Answer 1

$|x-1{|}^{{\ln }^{2}x-\ln x^2}=|x-1{|}^3$

$\Rightarrow {\ln }^{2}x-\ln x^2=3$, equate the exponents

$\Rightarrow {\ln }^{2}x-2\ln x-3=0$

$\Rightarrow (\ln x+1)(\ln x-3)=0$

$\Rightarrow \ln x=-1,3$

$\Rightarrow x=e^{-1},e^3\Rightarrow$ Two solutions

Note: $x=1$ is not in the domain of given equation