# What is the equation of the normal line of f(x)=xln(3^(1/x)) at x=0?

Dec 22, 2017

$x = 0$

#### Explanation:

$f \left(x\right) = x \ln \left({3}^{\frac{1}{x}}\right) = \ln \left({3}^{x \cdot \frac{1}{x}}\right) = \ln \left({3}^{1}\right) = \ln \left(3\right)$ (power rule for logarithms)

since $\ln \left(3\right)$ is a constant, $f \left(x\right)$ is a horizontal line. therefore the line normal to $f \left(x\right)$ must be vertical, so it follows the format: $x = a$

since the line is normal to $f \left(x\right)$ at x=0, 0=a.

so the normal line is simply: $x = 0$