Wind profile below the aerodynamic roughness length
The key text here is "for $z&gt;z_0$".  It's telling you that, while you can evaluate the equation for other values of $z$, outside of that range the equation is not a valid description of the physical system.  The equation could be written piece-wise to be complete:

$u(z) = \begin{cases} (u_*/k) \ln(z/z_0)&amp; z&gt;z_0 \\ 0 &amp; z\le z_0\end{cases}$

But this doesn't really add anything useful.  In practice, the "log-law" is used to describe the wind profile over 10s of metres and values of $z_0$ range from 1 mm to 2 m, so values of $z$ are likely to be in the valid region.  If you do need to make calculations that close to the surface (in the interfacial sublayer) then you'll need a different equation anyway.