Question

Решить уравнение.....

Answer 1

$2(log_5(2) - 1) + log_5(5^{\sqrt{x}} + 1) = log_5(5^{1 - \sqrt{x}} + 5) \\ 2log_5(\frac{2}{5} + log_5(5^{\sqrt{x}} + 1) = log_5(5^{-\sqrt{x}} + 1) + 1 \\ log_5(\frac{5^{\sqrt{x}} + 1}{5^{-\sqrt{x}} + 1}) = log_5(\frac{125}{4}) \\ t = 5^{\sqrt{x}} \\ 4(t+1) = 125(\frac{1}{t} + 1) \\ 4t^{2} - 121t - 125 = 0 \\ t_1 = \frac{121 + 129}{8} = \frac{125}{4} \\ t_2 = \frac{121 - 129}{8} (t > 0) \\ \sqrt{x} = log_5(\frac{125}{4}) \\ x = log_5^{2}(\frac{125}{4})$