log(sqrt(3))*log(sqrt(2))*(x-log(5)*1/log(6))

$$log{left (sqrt{2} right )} log{left (sqrt{3} right )} left(x – frac{log{left (5 right )}}{log{left (6 right )}}right) < 4$$

Дано неравенство:
$$log{left (sqrt{2} right )} log{left (sqrt{3} right )} left(x – frac{log{left (5 right )}}{log{left (6 right )}}right) < 4$$
Чтобы решить это нер-во – надо сначала решить соотвествующее ур-ние:
$$log{left (sqrt{2} right )} log{left (sqrt{3} right )} left(x – frac{log{left (5 right )}}{log{left (6 right )}}right) = 4$$
Решаем:
Дано уравнение:

log(sqrt(3))*log(sqrt(2))*(x-log(5)*1/log(6)) = 4

Раскрываем выражения:

x*log(2)*log(3)/4 – log(2)*log(3)*log(5)/(4*log(6)) = 4

Сокращаем, получаем:

-4 + x*log(2)*log(3)/4 – log(2)*log(3)*log(5)/(4*log(6)) = 0

Раскрываем скобочки в левой части ур-ния

-4 + x*log2log3/4 – log2log3log54*log+6) = 0

Переносим свободные слагаемые (без x)
из левой части в правую, получим:

x*log(2)*log(3) log(2)*log(3)*log(5)
————— – ——————– = 4
4 1
4*log (6)

Разделим обе части ур-ния на (x*log(2)*log(3)/4 – log(2)*log(3)*log(5)/(4*log(6)))/x

x = 4 / ((x*log(2)*log(3)/4 – log(2)*log(3)*log(5)/(4*log(6)))/x)

Получим ответ: x = log((2821109907456*5^log(3^log(2)))^(1/(log(2)*log(3)*log(6))))
$$x_{1} = log{left (left(2821109907456 cdot 5^{log{left (3^{log{left (2 right )}} right )}}right)^{frac{1}{log{left (2 right )} log{left (3 right )} log{left (6 right )}}} right )}$$
$$x_{1} = log{left (left(2821109907456 cdot 5^{log{left (3^{log{left (2 right )}} right )}}right)^{frac{1}{log{left (2 right )} log{left (3 right )} log{left (6 right )}}} right )}$$
Данные корни
$$x_{1} = log{left (left(2821109907456 cdot 5^{log{left (3^{log{left (2 right )}} right )}}right)^{frac{1}{log{left (2 right )} log{left (3 right )} log{left (6 right )}}} right )}$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} < x_{1}$$
Возьмём например точку
$$x_{0} = x_{1} – frac{1}{10}$$
=
$$- frac{1}{10} + log{left (left(2821109907456 cdot 5^{log{left (3^{log{left (2 right )}} right )}}right)^{frac{1}{log{left (2 right )} log{left (3 right )} log{left (6 right )}}} right )}$$
=
$$- frac{1}{10} + log{left (left(2821109907456 cdot 5^{log{left (3^{log{left (2 right )}} right )}}right)^{frac{1}{log{left (2 right )} log{left (3 right )} log{left (6 right )}}} right )}$$
подставляем в выражение
$$log{left (sqrt{2} right )} log{left (sqrt{3} right )} left(x – frac{log{left (5 right )}}{log{left (6 right )}}right) < 4$$

/ / 1
| | ———————–| |
| | 1 1 1 | |
| | log (2)*log (3)*log (6)| |
| |/ / log(2) | |
/ ___ / ___ | || log3 /| | 1 log(5)|
log/ 3 /*log/ 2 /*|log2821109907456*5 / / – — – ——-| < 4 | 10 1 | log (6)/

/ / 1
| | ——————–||
| | log(2)*log(3)*log(6)||
| |/ / log(2) || < 4 | 1 log(5) || log3 /| || / ___ / ___ |- -- - ------ + log2821109907456*5 / /|*log/ 2 /*log/ 3 / 10 log(6) /

значит решение неравенства будет при:
$$x < log{left (left(2821109907456 cdot 5^{log{left (3^{log{left (2 right )}} right )}}right)^{frac{1}{log{left (2 right )} log{left (3 right )} log{left (6 right )}}} right )}$$

_____
——-ο——-
x1

/ log(5) 16
And|-oo < x, x < ------ + -------------| log(6) log(2)*log(3)/

$$-infty < x wedge x < frac{log{left (5 right )}}{log{left (6 right )}} + frac{16}{log{left (2 right )} log{left (3 right )}}$$

/ 1
| ——————–|
| log(2)*log(3)*log(6)|
|/ / log(2) |
|| log3 /| |
(-oo, log2821109907456*5 / /)

$$x in left(-infty, log{left (left(2821109907456 cdot 5^{log{left (3^{log{left (2 right )}} right )}}right)^{frac{1}{log{left (2 right )} log{left (3 right )} log{left (6 right )}}} right )}right)$$