If the third term in the binomial expansion of

$\left(1+x^{\log _{2} x}\right)^{5}$

$\mathrm{T}_{3}={ }^{5} \mathrm{C}_{2} \cdot\left(\mathrm{x}^{\log _{2} \mathrm{x}}\right)^{2}=2560$

$\Rightarrow 10 \cdot x^{2 \log _{2} x}=2560$

$\Rightarrow x^{2 \log _{2} x}=256$

$\Rightarrow 2\left(\log _{2} x\right)^{2}=\log _{2} 256$

$\Rightarrow 2\left(\log _{2} x\right)^{2}=8$

$\Rightarrow\left(\log _{2} x\right)^{2}=4$

$\Rightarrow \quad \log _{2} x=2$ or $-2$

$x=4$ or $\frac{1}{4}$