51. (10-11 grades) Solve the equation $4^{-\frac{1}{x}}+10^{-\frac{1}{x}}=25^{-\frac{1}{x}}$.
51. Divide both parts of the equation by $\left(\frac{1}{25}\right)^{\frac{1}{x}}$. With respect to $\left(\frac{5}{2}\right)^{\frac{1}{x}}$, we obtain the quadratic equation $y^{2}+y-1=0$. Using the equality $\left(\frac{5}{2}\right)^{\frac{1}{x}}=y$, we find the value

$$
x=\frac{\lg 5-\lg 2}{\lg (\sqrt{5}-1)-\lg 2}
$$