Question

The equation $\sqrt{1+{\log }_x\sqrt{27}}{\log }_{3}x+1=0$ has which type of solution(s)?

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

Answer 1

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

The correct options are
A no integral solution
B one irrational solution
C no prime solution
D two real solutions

$\sqrt{1+{\log }_x\sqrt{27}}{\log }_{3}x+1=0$
Above equation is valid when $x>0$ and $x\ne 1$
$1+{\log }_x\sqrt{27}\ge 0-------------------------------1$
$\sqrt{1+{\log }_x\sqrt{27}}{\log }_{3}x+1=0$
$\Rightarrow \sqrt{1+\frac{3}{2{\log }_{3}x}}{\log }_{3}x=-1[\because \log a^m=m\log a\text{\&}{\log }_{b}a=\frac{\log a}{\log b}]$
Substitute $y={\log }_{3}x$ and squaring on both sides, we get
$y^2(1+\frac{3}{2y})=1$
$\Rightarrow 2y^2+3y-2=0$
$\Rightarrow y=-2,\frac{1}{2}$
$\Rightarrow {\log }_{3}x=-2,\frac{1}{2}$
$\Rightarrow x=\frac{1}{9},\sqrt{3}$

The values of x satisfy equation 1.

Ans: A,B,C,D