Question

The set of real values of x satisfying the equation
${|x-1|}^{lo{g}_3\left(x^2\right)-2lo{g}_x\left(9\right)}=(x-1{)}^7$
is

• A. {2}
  
• B. {27, 81}
  
• C. {81}
  
• D. {2, 81}

Answer 1

The correct option is C

{81}

${|x-1|}^{lo{g}_3\left(x^2\right)-2lo{g}_x\left(9\right)}=(x-1{)}^7(lo{g}_3{x}^2-2lo{g}_{x}9)lo{g}_{|x-1|}|x-1|=7lo{g}_{|x-1|}(x-1)lo{g}_3{x}^2-2lo{g}_{x}9=72lo{g}_{3}x-4\times \frac{1}{lo{g}_{3}x}=7Letlo{g}_{3}x=t2t-\frac{4}{t}=72t^2-7t-4=0(t-4\left)\right(2t+1)=0t=4,t=\frac{-1}{2}lo{g}_{3}x=4lo{g}_{3}x=\frac{-1}{2}x=81,x=\frac{1}{\sqrt{3}}<1\therefore x\ne \frac{1}{\sqrt{3}}$