5^x+5^(-x)=7

$$5^{x} + 5^{- x} = 7$$

Дано уравнение:
$$5^{x} + 5^{- x} = 7$$
или
$$5^{x} + 5^{- x} – 7 = 0$$
Сделаем замену
$$v = left(frac{1}{5}right)^{x}$$
получим
$$v – 7 + frac{1}{v} = 0$$
или
$$v – 7 + frac{1}{v} = 0$$
делаем обратную замену
$$left(frac{1}{5}right)^{x} = v$$
или
$$x = – frac{log{left (v right )}}{log{left (5 right )}}$$
Тогда, окончательный ответ
$$x_{1} = frac{1}{log{left (frac{1}{5} right )}} log{left (frac{1}{log{left (5 right )}} left(- log{left (2 right )} + log{left (3 sqrt{5} + 7 right )}right) right )} = frac{1}{log{left (5 right )}} left(- log{left (- log{left (2 right )} + log{left (3 sqrt{5} + 7 right )} right )} + log{left (log{left (5 right )} right )}right)$$
$$x_{2} = frac{1}{log{left (frac{1}{5} right )}} log{left (frac{1}{log{left (5 right )}} left(log{left (- 3 sqrt{5} + 7 right )} – log{left (2 right )}right) right )} = frac{1}{log{left (5 right )}} left(- log{left (log{left (2 right )} – log{left (- 3 sqrt{5} + 7 right )} right )} + log{left (log{left (5 right )} right )} – i piright)$$

$$x_{1} = frac{1}{log{left (5 right )}} left(- log{left (2 right )} + log{left (3 sqrt{5} + 7 right )}right)$$

/ ___
-log(2) + log7 – 3*/ 5 /
x2 = ————————–
log(5)

$$x_{2} = frac{1}{log{left (5 right )}} left(log{left (- 3 sqrt{5} + 7 right )} – log{left (2 right )}right)$$

x1 = -1.19597487133000

x2 = 1.19597487133000