Question

If $\alpha$ and $\beta$ are the roots of the equation $\left(\log _2 x\right)^2+4\left(\log _2 x\right)-1=0$ then the value of $\log _\beta \alpha+\log _\alpha \beta$ equals

• A. 18
• B. -16
• C. 14
• D. -18

Answer 1

Answer: (D)

$\log _2 \alpha+\log _2 \beta=-4 ; \log _2 \alpha \cdot \log _2 \beta=-1$
now $\log _\beta \alpha+\log _\alpha \beta=\frac{\log _2 \alpha}{\log _2 \beta}+\frac{\log _2 \beta}{\log _2 \alpha}=\frac{\left(\log _2 \alpha\right)^2+\left(\log _2 \beta\right)^2}{\log _2 \alpha \cdot \log _2 \beta}$
$=-\left[\left(\log _2 \alpha+\log _2 \beta\right)^2-2 \log _2 \alpha \cdot \log _2 \beta\right]$
$=-[16+2]=-18$.