uniqueness of solutions for laplace equation

Question

I know that the solutions to the Laplace equation $\triangle u=0$ are unique but in some cases one can find that there is more than 1 solution. For example $u=0$ and ln$|\textbf{x}|$ are both solutions to $\triangle u=0, u=0$ on $|\textbf{x}|=1$. How does this not contradict uniqueness?