Is the following criterion:
$$ \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial^2 f}{\partial y\partial x} $$
Equivalent to:
$$ \frac{\partial^2 \ln f}{\partial x\partial y} = \frac{\partial^2 \ln f}{\partial y\partial x} $$
Because $\ln$ function is a bijection?