Problem 1. Find $x$ for which

$$
\log _{2}\left(x^{2}+4\right)-\log _{2} x+x^{2}-4 x+2=0
$$
Solution. We have $x>0$ and the equation can be equivalently written as

$$
\log _{2}\left(x+\frac{4}{x}\right)=2-(x-2)^{2}
$$

Since $x+\frac{4}{x} \geq 2 \sqrt{x \cdot \frac{4}{x}}=4$, we obtain $\log _{2}\left(x+\frac{4}{x}\right) \geq 2 \ldots \ldots \ldots \ldots \ldots \ldots$. 2 p

![](https://cdn.mathpix.com/cropped/2024_06_07_179b284d1269acc77870g-1.jpg?height=49&width=1301&top_left_y=1060&top_left_x=434)

![](https://cdn.mathpix.com/cropped/2024_06_07_179b284d1269acc77870g-1.jpg?height=54&width=1304&top_left_y=1106&top_left_x=432)