2^((x*3)^(1/x))>6

$$2^{left(3 xright)^{frac{1}{x}}} > 6$$

Дано неравенство:
$$2^{left(3 xright)^{frac{1}{x}}} > 6$$
Чтобы решить это нер-во – надо сначала решить соотвествующее ур-ние:
$$2^{left(3 xright)^{frac{1}{x}}} = 6$$
Решаем:
$$x_{1} = frac{{Lambertw}{left (- frac{1}{3} log{left (log{left (6 right )} right )} + frac{1}{3} log{left (log{left (2 right )} right )} right )}}{- log{left (log{left (6 right )} right )} + log{left (log{left (2 right )} right )}}$$
$$x_{1} = frac{{Lambertw}{left (- frac{1}{3} log{left (log{left (6 right )} right )} + frac{1}{3} log{left (log{left (2 right )} right )} right )}}{- log{left (log{left (6 right )} right )} + log{left (log{left (2 right )} right )}}$$
Данные корни
$$x_{1} = frac{{Lambertw}{left (- frac{1}{3} log{left (log{left (6 right )} right )} + frac{1}{3} log{left (log{left (2 right )} right )} right )}}{- log{left (log{left (6 right )} right )} + log{left (log{left (2 right )} right )}}$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} < x_{1}$$
Возьмём например точку
$$x_{0} = x_{1} – frac{1}{10}$$
=

/ log(log(6)) log(log(2))
LambertW|- ———– + ———–|
3 3 / 1
————————————- – —
1 10
(-log(log(6)) + log(log(2)))

=
$$- frac{1}{10} + frac{{Lambertw}{left (- frac{1}{3} log{left (log{left (6 right )} right )} + frac{1}{3} log{left (log{left (2 right )} right )} right )}}{- log{left (log{left (6 right )} right )} + log{left (log{left (2 right )} right )}}$$
подставляем в выражение
$$2^{left(3 xright)^{frac{1}{x}}} > 6$$

/ 1
| ——————————————|
| / log(log(6)) log(log(2)) |
| LambertW|- ———– + ———–| |
| 3 3 / 1 |
| ————————————- – –|
| 1 10|
| (-log(log(6)) + log(log(2))) |
|// / log(log(6)) log(log(2)) |
|||LambertW|- ———– + ———–| | | |
||| 3 3 / 1 | | |
|||————————————- – –|*3| |
||| 1 10| | |
(-log(log(6)) + log(log(2))) / / /
2 > 6

/ 1
| ——————————————–|
| / log(log(6)) log(log(2))|
| LambertW|- ———– + ———–||
| 1 3 3 /|
| – — + ————————————-|
| 10 -log(log(6)) + log(log(2)) | > 6
|/ / log(log(6)) log(log(2)) |
|| 3*LambertW|- ———– + ———–|| |
|| 3 3 3 /| |
||- — + —————————————| |
10 -log(log(6)) + log(log(2)) / /
2

Тогда
$$x < frac{{Lambertw}{left (- frac{1}{3} log{left (log{left (6 right )} right )} + frac{1}{3} log{left (log{left (2 right )} right )} right )}}{- log{left (log{left (6 right )} right )} + log{left (log{left (2 right )} right )}}$$
не выполняется
значит решение неравенства будет при:
$$x > frac{{Lambertw}{left (- frac{1}{3} log{left (log{left (6 right )} right )} + frac{1}{3} log{left (log{left (2 right )} right )} right )}}{- log{left (log{left (6 right )} right )} + log{left (log{left (2 right )} right )}}$$

_____
/
——-ο——-
x1

$$x < infty wedge frac{{Lambertw}{left (- frac{1}{3} log{left (log{left (6 right )} right )} + frac{1}{3} log{left (log{left (2 right )} right )} right )}}{- log{left (log{left (6 right )} right )} + log{left (log{left (2 right )} right )}} < x$$

/ log(log(6)) log(log(2))
LambertW|- ———– + ———–|
3 3 /
(————————————-, oo)
-log(log(6)) + log(log(2))

$$x in left(frac{{Lambertw}{left (- frac{1}{3} log{left (log{left (6 right )} right )} + frac{1}{3} log{left (log{left (2 right )} right )} right )}}{- log{left (log{left (6 right )} right )} + log{left (log{left (2 right )} right )}}, inftyright)$$