Question: Solve for $n$. $\varphi = n[\ln(\alpha) - \ln(x)]$, where $\varphi = \frac{\partial [\ln(x)]}{\partial [\ln(y)]}$.
Solution:
Eq. 1: $\varphi = n[\ln(\alpha) - \ln(x)]$
Substitute $\varphi$ from Eq. 1 to Eq. 2:
Eq. 2: $\frac{\partial [\ln(x)]}{\partial [\ln(y)]} = n[\ln(\alpha) - \ln(x)]$
I try to equate the expression with $n$ to the expression of $\varphi$ then simplify it to $n = \frac{\partial [\ln(y)]}{\partial [\ln(x)](\ln(\alpha) - \ln(x))}$. However, the equation does not make sense because there are partial derivatives, should I solve the partial derivatives first before I substitute?
Any help is highly appreciated. Thank You.