352. Concerning graphics. For what value of $a$ does the graph of the function $a^{x}$ touch the graph of the function $\log _{a} x ?$
352. The function $y=\log _{a} x(a>0, a \neq 1, x>0)$ is the inverse of the function $y=a^{x}$. Due to the symmetry of their graphs with respect to the line $y=x$, we conclude that in the case of tangency, both graphs either touch the line $y=x$ or are perpendicular to it. Therefore, at the point of tangency, the following relations hold:

$$
y^{\prime}=a^{x} \ln a= \pm 1, \quad x=a^{x}
$$

The solutions to these equations will be $a=e^{\frac{1}{e}}(x=e)$ for the sign “+” and $a^{-e}\left(x=e^{-1}\right)$ for the sign “-”. Thus, the graph of $y=a^{x}$ is tangent to the graph of $y=\log _{a} x$ when $a=e^{\frac{1}{e}}$ and $e^{-e}$.

[V. Konechny, M. M., 45, 234 (March 1972).]