Theory of Equations 1 Question 30
31. The equation $x^{\frac{3}{4}\left(\log _2 x\right)^{2}+\log _2 x-\frac{5}{4}}=\sqrt{2}$ has
(1989; 2M)
(a) atleast one real solution
(b) exactly three real solutions
(c) exactly one irrational solution
(d) complex roots
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Solution:
- Given, $x^{\frac{3}{4}\left(\log _2 x\right)^{2}+\log _2 x-\frac{5}{4}}=\sqrt{2}$
$\Rightarrow \quad \frac{3}{4}\left(\log _2 x\right)^{2}+\log _2 x-\frac{5}{4}=\log _x \sqrt{2}$
$\Rightarrow \quad \frac{3}{4}\left(\log _2 x\right)^{2}+\log _2 x-\frac{5}{4}=\frac{1}{2 \log _2 x}$
$\Rightarrow \quad 3\left(\log _2 x\right)^{3}+4\left(\log _2 x\right)^{2}-5\left(\log _2 x\right)-2=0$
Put $\quad \log _2 x=y$
$\therefore \quad 3 y^{3}+4 y^{2}-5 y-2=0$
$\Rightarrow \quad(y-1)(y+2)(3 y+1)=0$
$\Rightarrow \quad y=1,-2,-1 / 3$
$\Rightarrow \quad \log _2 x=1,-2,-1 / 3$
$\Rightarrow \quad x=2, \frac{1}{2^{1 / 3}}, \frac{1}{4}$
