Converting derivative of log to one without log using chain rule

Question

If I have this:

$$ \frac{dlnx}{dln\eta} $$

How can I pull the log out of this derivative using the chain rule?

Answer 1

$$\frac{d\log[x]}{d\log[\eta]} = \frac{\frac{d\log[x]}{dx}}{\frac{d\log[\eta]}{d x}} = \frac{\frac{1}{x}}{\frac{d\log[\eta]}{d\eta}\frac{d\eta}{dx}} = \frac{\eta}{x}\frac{dx}{d\eta}$$