B2. Find all real numbers $x$ that satisfy the equation

$$
\sqrt{\log _{2} x}-\log _{2}(\sqrt{2} x)+\frac{5}{2}=0
$$

## 62nd Mathematical Competition for High School Students in Slovenia

Selection Competition, March 15, 2018

## Problems for 4th Year

## Time for solving: 45 minutes.
B2. Considering the relationship $\log _{2}(\sqrt{2} x)=\log _{2} \sqrt{2}+\log _{2} x=\frac{1}{2}+\log _{2} x$, we transform the given equation into

$$
\sqrt{\log _{2} x}-\log _{2} x+2=0
$$

We introduce a new variable $t=\sqrt{\log _{2} x}$, to obtain the quadratic equation

$$
t-t^{2}+2=0
$$

We rearrange the equation to $t^{2}-t-2=0$ and factor the left side as $(t-2)(t+1)=0$, yielding the solutions $t=-1$ and $t=2$. The first solution is discarded because $t \geq 0$, and from the second solution we get the equation $\sqrt{\log _{2} x}=2$, from which we calculate $x=2^{4}=16$.

![](https://cdn.mathpix.com/cropped/2024_06_07_cfd2cf88ee04390f062cg-15.jpg?height=60&width=1576&top_left_y=821&top_left_x=277)

![](https://cdn.mathpix.com/cropped/2024_06_07_cfd2cf88ee04390f062cg-15.jpg?height=65&width=1579&top_left_y=867&top_left_x=273)

Transforming the equation into $\sqrt{\log _{2} x}-\log _{2} x+2=0 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ points

Introducing a new variable or recognizing that the equation is factorable . . . . . 3 points

Factoring the quadratic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 points

Solutions $t=-1$ and $t=2$ or $\sqrt{\log _{2} x}=-1$ and $\sqrt{\log _{2} x}=2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots$.

Recognizing that the solution $t=-1$ or $\sqrt{\log _{2} x}=-1$ is invalid ..................... 2 points

![](https://cdn.mathpix.com/cropped/2024_06_07_cfd2cf88ee04390f062cg-15.jpg?height=54&width=1576&top_left_y=1184&top_left_x=274)

## 62nd Mathematical Competition for High School Students in Slovenia

Selection Competition, March 15, 2018

Each mathematically correct and complete solution is worth 20 points, even if the solving process is different from the official solutions.

## Solutions for 4th Year