Raising both sides of differential equation Quickie: say you have the expression;
$$\frac{d\ln(x)}{d\ln(y)} = z$$
Is it legit to raise both sides by $e$ so it becomes
original question
$$\frac{dx}{dy} = e^z$$
edited because I forgot about math
$$ e^{\left( \dfrac{d \ln x}{d \ln y}\right)} = e^z$$

What you wrote is pure formality, I've never seen such a notation. But if want to follow all rules of differentiation and etc., you could have done this
$$
d \ln x = \frac {dx}x \\
d \ln y = \frac {dy}y \\
\frac {d \ln x}{d \ln y} = \frac {y\,dx}{x\,dy} = z \\
\frac {dx}{dy} = z\frac xy
$$
