Question

The equation $x^{\{\frac{3}{4}{\left({\log }_{2}x\right)}^2+{\log }_{2}x-\frac{5}{4}\}}=\sqrt{2}$ has

• A. Atleast one real solution
• B. Exactly three real solution
• C. Exactly one irrational solution
• D. Complex roots

Answer 1

Answer: (C)

Exactly one irrational solution

It is given that

$x^{\{\frac{3}{4}{\left({\log }_{2}x\right)}^2+{\log }_{2}x+\frac{5}{2}\}}=\sqrt{2}$

$\Rightarrow \frac{3}{4}{\left({\log }_{2}x\right)}^2+{\log }_{2}x-\frac{5}{4}={\log }_x\sqrt{2}$

$\Rightarrow \frac{3}{4}{\left({\log }_{2}x\right)}^2+{\log }_{2}x-\frac{5}{4}=\frac{1}{2{\log }_{2}x}$

$\Rightarrow 3{\left({\log }_{2}x\right)}^3+4{\left({\log }_{2}x\right)}^2-5{\left({\log }_{2}x\right)}^{}-2$

Put ${\log }_{2}x=y$

Therefore, $3y^3+4y^3-5y-2=0$

$\Rightarrow y=1,-2,-\frac{1}{3}$

$\Rightarrow {\log }_{2}x=1,-2,-\frac{1}{3}$

$\Rightarrow x=2,\frac{1}{2^{1/3}},\frac{1}{4}$