# Equivalence of criteria using logarithmic transformation

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?

• Your criterion assesses closedness of $df$, not exactness. It is necessary but not sufficient. – Mark Fantini Jul 15 '14 at 21:04
• I don't understand what "assessing exactness of $df$" means here. The form $df$ is exact because it is $df$; there is nothing else to say about this. – user147263 Jul 16 '14 at 4:56
• I have edited the question. – jlandercy Jul 16 '14 at 12:31