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Common Roots

The equation ...

Question

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

A

At least one real solution

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B

Exactly three real solution

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C

Exactly one irrational solution

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D

Complex roots

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Solution

The correct option is C Exactly one irrational solution
taking the logarithm on both sides of your equation we obatin
\\ $(\frac{3}{4} (\frac{I n x}{I n 2}) + (\frac{I n (x - \frac{5}{4})}{I n 2})) I n x = 1 / 2 I n 2$
and here you can take a numerical method with such a method we obtain
$x \approx 2.050105808$

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