What to do when we can't apply the implicit function theorem?

Question

Consider $f(x,y,z)=xy\exp^{xz}-z\ln{y}=0$. It is clear by the implicit function theorem that we can solve for $x=x(y,z)$ around $(0,1,0)$. But as

$$\frac{\partial f}{\partial y}(0,1,0)=\frac{\partial f}{\partial z}(0,1,0)=0$$

we cannot do the same for $y=y(x,z)$ and $z=(x,y)$. How can I verify if these cases are or not possible (since the theorem doens't imply impossibility for them)?

Answer 1

Great question. Here's a hint for one solution: What's the second degree Taylor polynomial for $f$ at the point in question? Using it, can you locally solve for either $y$ or $z$?