Question

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

• A. at least one real solution
• B. exactly three real solutions
• C. exactly one irrational solution
• D. All of these

Answer 1

Answer: (D)

All of these

$X^{\left(3/4\right)(\log X-5/4)}=\sqrt{2}$

Take log base 2 on both sides.

${\log }_2{X}^{\left(3/4\right)(\log X-5/4)}={\log }_2{2}^{1/2}$

$\Rightarrow \frac{3}{4}(\log X-\frac{5}{4}).\log X=\frac{1}{2}$

$\Rightarrow {\log }^{2}X-\frac{5}{4}.\log X-\frac{2}{3}=0$

$\Rightarrow 12{\log }^{2}X-15\log X-8=0$

$\Rightarrow \log X=\frac{15\pm \sqrt{225+384}}{24}=\frac{15\pm \sqrt{609}}{24}$

$\Rightarrow X=2^{\frac{15+\sqrt{609}}{24}},2^{\frac{15-\sqrt{609}}{24}}$

Hence there are two irrational solutions of the equation.