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

$$
y=x^{2}+x\left(\log _{2} 20-\log _{5} 8\right)-\log _{2} 5-9 \log _{5} 2
$$

do not exceed 6.
Solution. Let $a=\log _{2} 5$. Then the condition of the problem will turn into the inequality

$$
x^{2}+\left(a-\frac{3}{a}+2\right) x-\left(a+\frac{9}{a}+6\right) \leqslant 0
$$

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

Answer: $-12 .(\mathrm{C})$

![](https://cdn.mathpix.com/cropped/2024_05_06_a8c31eb257d724803ce6g-06.jpg?height=67&width=901&top_left_y=1848&top_left_x=629)