MCQ
The equation ${x^{(3/4){{({{\log }_2}x)}^2} + ({{\log }_2}x) - 5/4}} = \sqrt 2 $ has
- AAt least one real solution
- BExactly three real solutions
- CExactly one irrational solution
- ✓All the above
✓
Answer
Correct option: D.
All the above
d
(d) For the given equation to be meaningful we must have $x > 0$. For $x > 0$ the given equation can be written as $\frac{3}{4}{({\log _2}x)^2} + {\log _2}x - \frac{5}{4} = {\log _x}\sqrt 2 = \frac{1}{2}{\log _x}2$
(d) For the given equation to be meaningful we must have $x > 0$. For $x > 0$ the given equation can be written as $\frac{3}{4}{({\log _2}x)^2} + {\log _2}x - \frac{5}{4} = {\log _x}\sqrt 2 = \frac{1}{2}{\log _x}2$
.==> $\frac{3}{4}{t^2} + t - \frac{5}{4} = \frac{1}{2}\left( {\frac{1}{t}} \right)$
By putting $t = {\log _2}x$ so that ${\log _x}2 = \frac{1}{t}$ because
${\log _2}x{\log _x}2 = 1$.
==> $3{t^3} + 4{t^2} - 5t - 2 = 0\,\,\,\, \Rightarrow (t - 1)(t + 2)(3t + 1) = 0$
==> ${\log _2}x = t = 1, - 2, - \frac{1}{3}$
==> $x = 2,{2^{ - 2}},{2^{ - 1/3}}$or $x = 2,\frac{1}{4},\frac{1}{{{2^{1/3}}}}$
Thus the given equation has exactly three real solutions out of which exactly one is irrational namely $\frac{1}{{{2^{1/3}}}}$.
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