True or False: $f(z)=Ln(z)$ is periodic

Question

From Wolfram MathWorld:

A function $f(x)$ is said to be periodic with period $p$ if $f(x)=f(x+np)$ for $n=1,2,3...$

The easy bait here is to realize that $Ln(z)$ is asking for the principal value of the natural logarithm (due to the capital letter L), so my guess would be that no, this is not a periodic function because the domain of the graph of $Ln(z)$ is from $-\pi$ to $\pi$.