20*log(5*x^(9/4))

$$20 log{left (5 x^{frac{9}{4}} right )} leq frac{3 log{left (x right )}}{log{left (125 right )}} + frac{7 log{left (25 x right )}}{log{left (25 right )}}$$

Дано неравенство:
$$20 log{left (5 x^{frac{9}{4}} right )} leq frac{3 log{left (x right )}}{log{left (125 right )}} + frac{7 log{left (25 x right )}}{log{left (25 right )}}$$
Чтобы решить это нер-во – надо сначала решить соотвествующее ур-ние:
$$20 log{left (5 x^{frac{9}{4}} right )} = frac{3 log{left (x right )}}{log{left (125 right )}} + frac{7 log{left (25 x right )}}{log{left (25 right )}}$$
Решаем:
$$x_{1} = frac{5^{- frac{14}{- 90 log{left (5 right )} + 9}}}{e^{- frac{40 log^{2}{left (5 right )}}{- 90 log{left (5 right )} + 9}}}$$
$$x_{1} = frac{5^{- frac{14}{- 90 log{left (5 right )} + 9}}}{e^{- frac{40 log^{2}{left (5 right )}}{- 90 log{left (5 right )} + 9}}}$$
Данные корни
$$x_{1} = frac{5^{- frac{14}{- 90 log{left (5 right )} + 9}}}{e^{- frac{40 log^{2}{left (5 right )}}{- 90 log{left (5 right )} + 9}}}$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} leq x_{1}$$
Возьмём например точку
$$x_{0} = x_{1} – frac{1}{10}$$
=

2
-14 40*log (5)
—————- —————-
1 1
(9 – 90*log(5)) (9 – 90*log(5)) 1
5 *e – —
10

=
$$- frac{1}{10} + frac{5^{- frac{14}{- 90 log{left (5 right )} + 9}}}{e^{- frac{40 log^{2}{left (5 right )}}{- 90 log{left (5 right )} + 9}}}$$
подставляем в выражение
$$20 log{left (5 x^{frac{9}{4}} right )} leq frac{3 log{left (x right )}}{log{left (125 right )}} + frac{7 log{left (25 x right )}}{log{left (25 right )}}$$

/ 2 / / 2
/ 9/4 | -14 40*log (5) | | | -14 40*log (5) ||
| / 2 | | —————- —————- | | | —————- —————- ||
| | -14 40*log (5) | | | 1 1 | | | 1 1 ||
| | —————- —————- | | | (9 – 90*log(5)) (9 – 90*log(5)) 1 | | | (9 – 90*log(5)) (9 – 90*log(5)) 1 ||
| | 1 1 | | 3*log|5 *e – –| 7*log|25*|5 *e – –||
| | (9 – 90*log(5)) (9 – 90*log(5)) 1 | | 10/ 10//
20*log|5*|5 *e – –| | <= ----------------------------------------------- + ---------------------------------------------------- 10/ / 1 1 log (125) log (25)

/ 2 / 2
/ 9/4 | -14 40*log (5) | | -14 40*log (5) |
| / 2 | | ————- ————-| | ————- ————-|
| | -14 40*log (5) | | | 1 9 – 90*log(5) 9 – 90*log(5)| | 5 9 – 90*log(5) 9 – 90*log(5)|
| | ————- ————-| | <= 3*log|- -- + 5 *e | 7*log|- - + 25*5 *e | | | 1 9 - 90*log(5) 9 - 90*log(5)| | 10 / 2 / 20*log|5*|- -- + 5 *e | | ------------------------------------------- + --------------------------------------------- 10 / / log(125) log(25)

значит решение неравенства будет при:
$$x leq frac{5^{- frac{14}{- 90 log{left (5 right )} + 9}}}{e^{- frac{40 log^{2}{left (5 right )}}{- 90 log{left (5 right )} + 9}}}$$

_____
——-•——-
x1

$$x leq frac{5^{- frac{14}{- 90 log{left (5 right )} + 9}}}{e^{- frac{40 log^{2}{left (5 right )}}{- 90 log{left (5 right )} + 9}}} wedge -infty < x$$

2
-14 40*log (5)
————- ————-
9 – 90*log(5) 9 – 90*log(5)
(-oo, 5 *e ]

$$x in left(-infty, frac{5^{- frac{14}{- 90 log{left (5 right )} + 9}}}{e^{- frac{40 log^{2}{left (5 right )}}{- 90 log{left (5 right )} + 9}}}right]$$