One dimensional map

Question

I have the 1D map $x_{n+1}=Log(x_n)$. I have noticed that when an iterate becomes negative, the orbit becomes complex. Had never seen this before. What happens to the orbit that starts at some $x_0$ such that $x_n<0$ for some $n>0$, since in that case $x_{n+1}=Log(x_n)$ will be complex valued? Does the orbit terminate at the negative iterate? Thanks!