The two given functions, $x= \log y$ and $y= \log x$, or more generally, $x= f(y)$ and $y= f(x)$, just swap $(x, y)$ with $(y, x)$. The line of "reflection" if any, must be the perpendicular bisector of every line segment from $(x, y)$ with $(y, x)$.
In particular, the line from $(x_0, y_0)$ to $(y_0, x_0)$ has slope $\dfrac{x_0- y_0}{y_0- x_0}= -1$. Any line perpendicular to that must have slope 1. The line segement from $(x_0, y_0)$ to $(y_0, x_0)$ has midpoint $(\frac{x_0+ y_0}{2}, \frac{y_0+ x_0}{2})$. So the perpendicular bisector of that line segment is $y= 1\left(x- \frac{x_0+ y_0}{2}\right)+ \frac{x_0+ y_0}{2}$ which is simply $y= x$. Notice that this does NOT depend upon the specific point $(x_0, y_0)$.