Question

The equation $\sqrt{1+\log \sqrt{27}}{\log }_{2}x+1=0$ has

• A. no integral solution
• B. one irrational solution
• C. two real solutions
• D. no prime solution

Answer 1

Answer: (A), (D)

$\left(\sqrt{1+\frac{3}{2{\log }_{3}x}}\right){\log }_{3}x+1=0$
$\text{ let }{\log }_{3}x=y$
$\left(\sqrt{1+\frac{3}{2y}}\right)y=-1\Rightarrow \left(1+\frac{3}{2y}\right)=\frac{1}{y^2}\Rightarrow \frac{2y+3}{2y}=\frac{1}{y^2}$
$2y^2+3y-2=0\Rightarrow 2y^2+4y-y-2=0\Rightarrow (y+2)(2y-1)=0$
$y=1/2\text{ or }y=-2\Rightarrow x=3^{1/2}\text{ (rejected) or }x=1/9$