5. Solve the equation:

$$
\log \left(\frac{1}{x}\right) \cdot \log \left(\frac{4}{x}\right)=\frac{3}{4} \cdot(\log 4)^{2}
$$

Solve each problem independently on the attached paper, and each problem on its own page.

Do not sign the sheets, write only your code.

Write your answers with a pen clearly and neatly. Each problem's solution will be graded from 0 to 6 points.

You have 120 minutes to solve the problems.

THE NATIONAL COMPETITION COMMITTEE WISHES YOU GREAT SUCCESS.

4th national mathematics competition for students of secondary technical and vocational schools Maribor, April 17, 2004

## PROBLEMS FOR 4th GRADE
5. First, we use the rule for logarithmizing a quotient: $(\log 1-\log x) \cdot(\log 4-\log x)=$ $\frac{3}{4}(\log 4)^{2}$. We rearrange the equation to $(\log x)^{2}-\log x \cdot \log 4-\frac{3}{4}(\log 4)^{2}=0$. We calculate the discriminant of the quadratic equation $D=4(\log 4)^{2}$, which has solutions $(\log x)_{1}=\frac{3}{2} \log 4=\log 8$ and $(\log x)_{2}=$ $-\frac{1}{2} \log 4=\log \frac{1}{2}$. From these solutions, we get the solutions of the given equation: $x_{1}=8$ and $x_{2}=\frac{1}{2}$.

Using the rule for logarithmizing a quotient .............................................................................. Rearranged quadratic equation: $(\log x)^{2}-\log x \cdot \log 4-\frac{3}{4}(\log 4)^{2}=0 \ldots \ldots \ldots \ldots \ldots \ldots .1$ point

Calculated discriminant $D=4(\log 4)^{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Solutions of the quadratic equation: $(\log x)_{1}=\log 8,(\log x)_{2}=\log \frac{1}{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-11.jpg?height=69&width=1636&top_left_y=2464&top_left_x=210)

## Fourth Year