Question

The value of $\log {}_{(x+\frac{1}{x})}\left({\log }_2\frac{x-1}{x+2}\right)$ is defined if the value of x lies in the set

• A. $\left(0,\infty \right)$
• B. $[0,\infty )$
• C. $\left(1,\infty \right)$
• D. $\varphi$

Answer 1

The correct option is D $\varphi$

$\mathrm{Given}\mathrm{that},\log {}_{(x+\frac{1}{x})}\left({\log }_2\frac{x-1}{x+2}\right)\mathrm{Now},\left({\log }_2\frac{x-1}{x+2}\right)\mathrm{will}\mathrm{be}\mathrm{defined},\mathrm{if}\frac{x-1}{x+2}>0\Rightarrow x<-2\mathrm{or}x>1....\left(1\right]\log {}_{(x+\frac{1}{x})}\left({\log }_2\frac{x-1}{x+2}\right)\mathrm{will}\mathrm{be}\mathrm{defined},\mathrm{if}x+\frac{1}{x}>0\Rightarrow \frac{x^2+1}{x}>0\Rightarrow x>0...\left(2\right]\mathrm{Also},{\log }_2\frac{x-1}{x+2}>0\Rightarrow \frac{x-1}{x+2}>2^0=1\Rightarrow \frac{x-1}{x+2}-1>0\Rightarrow \frac{x-1-x-2}{x+2}>0\Rightarrow \frac{-3}{x+2}>0\Rightarrow x+2<0\Rightarrow x<-2...\left(3\right]\mathrm{From}\left(1\right],\left(2\right]\mathrm{and}\left(3\right],\mathrm{there}\mathrm{is}\mathrm{no}\mathrm{value}\mathrm{of}x\mathrm{for}\mathrm{which}\mathrm{the}\mathrm{given}\mathrm{expression}\mathrm{is}\mathrm{defined}.$