First slide

Theory of equations

Question

The equation $x^{(3 / 4) {(\log_{2} x)}^{2} + {(\log_{2} x)}^{- 5 / 4}} = \sqrt{2}$ has

Moderate

A

At least one real solution
B

Exactly three real solutions
C

Exactly one irrational solution
D

All the above

Solution

For the given equation to be meaningful we must have $x > 0$ . For $x > 0$ the given equation can be written as
$\frac{3}{4} {(\log_{2} x)}^{2} + \log_{2} x - \frac{5}{4} = \log_{x} \sqrt{2} = \frac{1}{2} \log_{x} 2$
$\Rightarrow \frac{3}{4} t^{2} + t - \frac{5}{4} = \frac{1}{2} (\frac{1}{t})$
By putting $t = \log_{2} x$ so that $\log_{x} 2 = \frac{1}{t}$ because
$\log_{2} x \log_{x} 2 = 1 . \Rightarrow 3 t^{3} + 4 t^{2} - 5 t - 2 = 0 \Rightarrow (t - 1) (t + 2) (3 t + 1) = 0 \Rightarrow \log_{2} x = t = 1, - 2, - \frac{1}{3} \Rightarrow x = 2, 2^{- 2}, 2^{- 1 / 3} or x = 2, \frac{1}{4}, \frac{1}{2^{1 / 3}}$
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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