5.4. Find the sum of all integer values of the argument $x$ for which the corresponding values of the function

$$
y=x^{2}+x\left(\log _{5} 2-\log _{2} 10\right)-\log _{2} 25-3 \log _{5} 2
$$

do not exceed 7.
Solution. Let $a=\log _{2} 5$. Then the condition of the problem will turn into the inequality

$$
x^{2}-\left(a-\frac{1}{a}+1\right) x-\left(2 a+\frac{3}{a}+7\right) \leqslant 0 .
$$

Considering that $a \in(2,3)$, we get $x \in\left[-\frac{1}{a}-2, a+3\right]$. Since $-3<-\frac{1}{a}-2<-2,5<a+3<6$, the integer solutions will be the numbers $-2,-1,0,1,2,3,4,5$.

Answer: 12. (D)

![](https://cdn.mathpix.com/cropped/2024_05_06_a8c31eb257d724803ce6g-07.jpg?height=68&width=784&top_left_y=1014&top_left_x=682)