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
$$ e^{\left( \dfrac{d \ln x}{d \ln y}\right)} = e^z$$

We have $df(x)=f'(x)dx$ so
$$z=\frac{d\ln x}{d\ln y}=\frac{y\,dx}{x\,dy}$$
hence
$$\frac{dx}{dy}=z\frac{x}{y}.$$
