log(3)^2*(x^2-16)-5*log(3)*(x^2-16)+6>=0

$$- 5 left(x^{2} – 16right) log{left (3 right )} + left(x^{2} – 16right) log^{2}{left (3 right )} + 6 geq 0$$

Дано неравенство:
$$- 5 left(x^{2} – 16right) log{left (3 right )} + left(x^{2} – 16right) log^{2}{left (3 right )} + 6 geq 0$$
Чтобы решить это нер-во – надо сначала решить соотвествующее ур-ние:
$$- 5 left(x^{2} – 16right) log{left (3 right )} + left(x^{2} – 16right) log^{2}{left (3 right )} + 6 = 0$$
Решаем:
Раскроем выражение в уравнении
$$- 5 left(x^{2} – 16right) log{left (3 right )} + left(x^{2} – 16right) log^{2}{left (3 right )} + 6 = 0$$
Получаем квадратное уравнение
$$- 5 x^{2} log{left (3 right )} + x^{2} log^{2}{left (3 right )} – 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )} = 0$$
Это уравнение вида

a*x^2 + b*x + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$x_{1} = frac{sqrt{D} – b}{2 a}$$
$$x_{2} = frac{- sqrt{D} – b}{2 a}$$
где D = b^2 – 4*a*c – это дискриминант.
Т.к.
$$a = – 5 log{left (3 right )} + log^{2}{left (3 right )}$$
$$b = 0$$
$$c = – 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}$$
, то

D = b^2 – 4 * a * c =

(0)^2 – 4 * (log(3)^2 – 5*log(3)) * (6 – 16*log(3)^2 + 80*log(3)) = -(-20*log(3) + 4*log(3)^2)*(6 – 16*log(3)^2 + 80*log(3))

Т.к. D > 0, то уравнение имеет два корня.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b – sqrt(D)) / (2*a)

или
$$x_{1} = frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
$$x_{2} = – frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
$$x_{1} = frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
$$x_{2} = – frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
$$x_{1} = frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
$$x_{2} = – frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
Данные корни
$$x_{1} = frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
$$x_{2} = – frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
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 / 2
/ – 4*log (3) + 20*log(3) */ 6 – 16*log (3) + 80*log(3) 1
———————————————————— – —
1 10
/ 2
-10*log(3) + 2*log (3)/

=
$$frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}} – frac{1}{10}$$
подставляем в выражение
$$- 5 left(x^{2} – 16right) log{left (3 right )} + left(x^{2} – 16right) log^{2}{left (3 right )} + 6 geq 0$$

/ 2 / 2
|/ _________________________ ____________________________ | |/ _________________________ ____________________________ |
|| / 2 / 2 | | || / 2 / 2 | |
2 ||/ – 4*log (3) + 20*log(3) */ 6 – 16*log (3) + 80*log(3) 1 | | ||/ – 4*log (3) + 20*log(3) */ 6 – 16*log (3) + 80*log(3) 1 | |
log (3)*||———————————————————— – –| – 16| – 5*log(3)*||———————————————————— – –| – 16| + 6 >= 0
|| 1 10| | || 1 10| |
|| / 2 | | || / 2 | |
-10*log(3) + 2*log (3)/ / / -10*log(3) + 2*log (3)/ / /

/ 2 / 2
| / _________________________ ____________________________ | | / _________________________ ____________________________ |
| | / 2 / 2 | | | | / 2 / 2 | |
2 | | 1 / – 4*log (3) + 20*log(3) */ 6 – 16*log (3) + 80*log(3) | | | | 1 / – 4*log (3) + 20*log(3) */ 6 – 16*log (3) + 80*log(3) | | >= 0
6 + log (3)*|-16 + |- — + ————————————————————| | – 5*|-16 + |- — + ————————————————————| |*log(3)
| | 10 2 | | | | 10 2 | |
-10*log(3) + 2*log (3) / / -10*log(3) + 2*log (3) / /

но

/ 2 / 2
| / _________________________ ____________________________ | | / _________________________ ____________________________ |
| | / 2 / 2 | | | | / 2 / 2 | |
2 | | 1 / – 4*log (3) + 20*log(3) */ 6 – 16*log (3) + 80*log(3) | | | | 1 / – 4*log (3) + 20*log(3) */ 6 – 16*log (3) + 80*log(3) | | < 0 6 + log (3)*|-16 + |- -- + ------------------------------------------------------------| | - 5*|-16 + |- -- + ------------------------------------------------------------| |*log(3) | | 10 2 | | | | 10 2 | | -10*log(3) + 2*log (3) / / -10*log(3) + 2*log (3) / /

Тогда
$$x leq frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$
не выполняется
значит одно из решений нашего неравенства будет при:
$$x geq frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}} wedge x leq – frac{sqrt{- 4 log^{2}{left (3 right )} + 20 log{left (3 right )}}}{- 10 log{left (3 right )} + 2 log^{2}{left (3 right )}} sqrt{- 16 log^{2}{left (3 right )} + 6 + 80 log{left (3 right )}}$$

_____
/
——-•——-•——-
x1 x2

/ ______________________________________________ ______________________________________________
| / 2 / 2 |
| / 6 – 16*log (3) + 2*log(12157665459056928801) -/ 6 – 16*log (3) + 2*log(12157665459056928801) |
And|x <= -------------------------------------------------, --------------------------------------------------- <= x| | ____________ ________ ____________ ________ | / 5 - log(3) */ log(3) / 5 - log(3) */ log(3) /

$$x leq frac{sqrt{- 16 log^{2}{left (3 right )} + 6 + 2 log{left (12157665459056928801 right )}}}{sqrt{- log{left (3 right )} + 5} sqrt{log{left (3 right )}}} wedge – frac{sqrt{- 16 log^{2}{left (3 right )} + 6 + 2 log{left (12157665459056928801 right )}}}{sqrt{- log{left (3 right )} + 5} sqrt{log{left (3 right )}}} leq x$$

______________________________________________ ______________________________________________
/ 2 / 2
-/ 6 – 16*log (3) + 2*log(12157665459056928801) / 6 – 16*log (3) + 2*log(12157665459056928801)
[—————————————————, ————————————————-]
____________ ________ ____________ ________
/ 5 – log(3) */ log(3) / 5 – log(3) */ log(3)

$$x in left[- frac{sqrt{- 16 log^{2}{left (3 right )} + 6 + 2 log{left (12157665459056928801 right )}}}{sqrt{- log{left (3 right )} + 5} sqrt{log{left (3 right )}}}, frac{sqrt{- 16 log^{2}{left (3 right )} + 6 + 2 log{left (12157665459056928801 right )}}}{sqrt{- log{left (3 right )} + 5} sqrt{log{left (3 right )}}}right]$$