13. If the equation $a^{x}=x(a>0, a \neq 1)$ has two distinct real roots, then the range of the real number $a$ is
Given $10$, we have $x \cdot \ln a - \ln x = 0 \Rightarrow \ln a = \frac{\ln x}{x}$. Let $f(x) = \frac{\ln x}{x} (x > 0)$, then $f'(x) = \frac{1 - \ln x}{x^2}$. When $x \in (0, e)$, $f'(x) > 0$; when $x \in (e, +\infty)$, $f'(x) < 0$, so $f(x)$ is increasing on $(0, e)$ and decreasing on $(e, +\infty)$. Therefore, $0 < \ln a < f(e) = \frac{1}{e}$, which means $1 < a < e^{\frac{1}{e}}$.