# What is the equation of the normal line of f(x)=xln(3^(1/x)) at x=9?

The normal line at $x = 9$ is the vertical line $x = 9$
$f \left(x\right) = x \left[\ln \left({3}^{\frac{1}{x}}\right)\right] = x \left[\frac{1}{x} \ln \left(3\right)\right] = \ln \left(3\right)$
So the graph of $f$ is a horizontal line and the normal line if vertical.
The vertical line with $x = 9$ has equation $x = 9$.