Question

The equation $x^{\frac{3}{4}(\log_2 x)^2+\log_2x-\frac{5}{4}} =\sqrt 2$ has

• A. atleast one real solution
• B. exactly three real solutions
• C. exactly one irrational solution
• D. complex roots

Answer 1

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

Given, $x^{\frac{3}{4}(\log_2 x)^2+\log_2x-\frac{5}{4}}=\sqrt2$
$\Rightarrow {\frac{3}{4}(\log_2 x)^2+\log_2x-\frac{5}{4}=\log_x\sqrt2}$
$\Rightarrow {\frac{3}{4}(\log_2 x)^2+\log_2x-\frac{5}{4}=\frac{1}{2\log_2x}}$
$\Rightarrow 3(\log_2 x)^3+4(\log_2x)^2-5(\log_2x)-2=0$
Put $ \log_2x=y$
$\therefore 3y^3+4y^2-5y-2=0$
$\Rightarrow (y-1)(y+2)(3y+1)=0$
$\Rightarrow y=1,-2,-1/3$
$\Rightarrow \log_2x=1,-2,-1/3$
$\Rightarrow x=2,\frac{1}{2^{1/3}},\frac{1}{4}$