Answer
$z$ IS a function of $x$ and $y$.
Work Step by Step
For $z$ to be a function of $x$ and $y$ then we require that for every ordered pair $(x,y)$ there is only one value of $z$ obtained from the formula connecting them.
Here we have
$$z+x\ln y - 8yz=0.$$
Solving for $z$ we get
$$z+x\ln y - 8yz=0\Rightarrow z(1-8y) = -x\ln y\Rightarrow z=\frac{x\ln y}{8y-1}.$$
Here we see that for each ordered pair $(x,y)$ we get exactly one value for $z$ so $z$ IS a function of $x$ and $y$.
