Answer
As $\nabla \times \textbf F \ne 0$, so, $\textbf F$ is not conservative
Work Step by Step
$\textbf F(x,y) = y\ln z \textbf i - x\ln z \textbf j+\frac{xy}{z} $
$\nabla \times \textbf F =\begin{vmatrix}
\textbf i & \textbf j & \textbf k\\
\frac{\delta}{\delta x} & \frac{\delta}{\delta y}& \frac{\delta}{\delta z} \\
y\ln z & - x\ln z & \frac{xy}{z} \\
\end{vmatrix} = \frac{2x}{z}\textbf i + 0\textbf j -2\ln z \textbf k \ne 0
\\$
As $\nabla \times \textbf F \ne 0$, so, $\textbf F$ is not conservative
