Question

If ${\left(x-1\right|}^{{\left({\log }_{10}x\right)}^2-{\log }_{10}{x}^2}={\left(x-1\right|}^3$ , then the value/s of x is/are

• A. 1000
• B. 2
• C. 0.1
• D. 100

Answer 1

Answer: (A), (B), (C)

0.1

We have, ${\left(x-1\right|}^{{\left({\log }_{10}x\right)}^2-{\log }_{10}{x}^2}={\left(x-1\right|}^3,x>0,x\ne 1\Rightarrow \left(x-1\right|=1\mathrm{or}{\left({\log }_{10}x\right)}^2-{\log }_{10}{x}^2=3\Rightarrow x=2\mathrm{or}{\left({\log }_{10}x\right)}^2-2{\log }_{10}x-3=0\Rightarrow x=2\mathrm{or}\left(\left({\log }_{10}x-3\right)\left({\log }_{10}x+1\right)\right]=0\Rightarrow x=2\mathrm{or}{\log }_{10}x=3,{\log }_{10}x=-1\Rightarrow x=2,x={10}^3=1000,x={10}^{-1}=0.1$