Question

The equation, $\log _2\left(2 x^2\right)+\log _2 x \cdot x^{\log _x\left(\log _2 x+1\right)}+\frac{1}{2} \log _4{ }^2 x^4+2^{-3 \log _{\nu 2}\left(\log _2 x\right)}=1$ has :

• A. exactly one real solution
• B. two real solutions
• C. 3 real solutions
• D. no solution.

Answer 1

Answer: (D)

$ y^3+3 y^2+3 y+1=1 \text { where } y=\log _2 x \Rightarrow y\left(y^2+3 y+3\right)=0 $
$\Rightarrow x=0 \text { (rejected) } ; y^2+3 y+3 \text { (complex roots) }$