4. Solve the equation $\left(x^{\log _{2} x}-1\right)^{2}=1$.
Solution. We have by row:

$$
\begin{aligned}
& x^{2 \log _{2} x}-2 x^{\log _{2} x}+1=1 \\
& x^{2 \log _{2} x}=2 x^{\log _{2} x} \\
& 2\left(\log _{2} x\right)^{2}=\log _{2} 2+\left(\log _{2} x\right)^{2} \\
& \left(\log _{2} x\right)^{2}=1 \\
& \log _{2} x= \pm 1 \\
& x_{1}=\frac{1}{2} \text { and } x_{2}=2
\end{aligned}
$$

## 4th year