I think, it would be good to rethink, what cylinder coordinates are, before going on. There seem to be a lot of misconceptions in your mind.
First of all you should clearly distinguish between vectors and components of vectors with respect to a basis. To introduce cylinder coordinates one usually starts with cartesian coordinates. The position vector is given by
\vec{r}=x \vec{i} + y \vec{j} + z \vec{k}.
Here, (\vec{i},\vec{j},\vec{k}) are a set of three constant unit vectors that are perpendicular to each other and build a right-handed dreibein, i.e., they fulfill
\vec{k}=\vec{i} \times \vec{j}.
Then you introduce cylinder coordinates by setting
\vec{r}=\rho \cos \theta \vec{i} + \rho \sin \theta \vec{j} + z \vec{k}.
I use \rho instead of r, because usually r=|\vec{r}|=\sqrt{\rho^2+z^2}\neq \rho.
The ranges of the coordinates are \rho>0, \theta \in [0,2\pi), z \in \mathbb{R}. This covers the entire space except the z axis, where the cylindrical coordinates are singular.
Now, you introduce also a new basis, adapted to the new coordinates. The new basis is defined by the tangent vectors to the coordinate lines. If these tangent vectors are perpendicular at any point in the domain of the new coordinates, you call the coordinates "orthogonal curvilinear coordinates" and then you use a normalized basis. Then you have again a orthonormal basis, but in general these basis vectors will not be constant.
In the case of cylinder coordinates the new basis vectors are
\hat{\rho}=\frac{\partial_{\rho} \vec{r}}{|\partial_{\rho} \vec{r}|}=\cos \theta \vec{i} + \sin \theta \vec{j}, \\<br />
\hat{\theta}=\hat{\rho}=\frac{\partial_{\theta} \vec{r}}{|\partial_{\theta} \vec{r}|}=-\sin \theta \vec{i}+\cos \theta \vec{j},\\<br />
\hat{z}=\frac{\partial_{z} \vec{r}}{|\partial_{z} \vec{r}|}=\vec{k}.
Obviously these vectors are perpendicular to each other, and also build a right-handed system at each point (except along the z axis where the cylinder coordinates are singular).
Now you write the vector fields in terms of the components wrt. the new basis, i.e.,
\vec{V}=V_r \hat{r} + V_{\theta} \hat{\theta} + V_z \hat{z}.
For your case you need \vec{j} in terms of the new coordinates. Since you have orthonormal coordinates, the components are easily calculated by taking the dot products with the new basis, i.e., you have
\vec{j} \cdot \hat{r}=\sin \theta, \quad \vec{j} \cdot \hat{\theta}=\cos \theta, \quad \vec{j} \cdot \hat{z}=0.
Thus you have
\vec{j}=\sin \theta \hat{r} + \cos \theta \hat{\theta},
and you vector field, expressed in cylinder coordinates, is
\vec{V}=\ln(\rho \sin \theta) (\sin \theta \hat{r} + \cos \theta \hat{\theta}).
Now you can take the curl directly in cylinder coordinates, using the corresponding formula from your textbook.
That's, however, not very convenient, because it's easier to take the curl in Cartesian coordinates in this case, because obviously
\vec{V}=\vec{j} \ln y.
Further, since \vec{j}=\text{const} you have
\vec{\nabla} \times \vec{V} =-\vec{j} \times \vec{\nabla} \ln y.
You should calculate the curl in both ways and prove that you indeed get the same result!
