How do I simplify a Multivariable expression involving derivatives of logarithms?

Question

I have this expression I got after a lot of calculation:

$$\sigma =\frac{d\log\left(\frac{b(x,y,\rho)}{r(x,y,\rho)}\right)}{d\log\left(\frac{ 2 }{3}\left(\frac{b(x,y,\rho)}{r(x,y,\rho)}\right)^{-\rho-1}\right)}$$

I know $$[\log(f)]' = \frac{f'}{f}$$ but the problem is, I am uncertain how to handle this since $b$ and $r$ are multivariable.

My Question

I want to simplify the logs. Will applying total derivatives help somehow? Or am I overlooking some obvious simplification.

Answer 1

Let $u= \log\frac{b(x,y,\rho)}{r(x,y,\rho)}$, so that $\log\left(\frac{ 2 }{3}\left(\frac{b(x,y,\rho)}{r(x,y,\rho)}\right)^{-\rho-1}\right)=\log\frac{2}{3} - (\rho +1)u.$

Then,

$$\sigma =\frac{d u}{d\left[\log\frac{2}{3} - (\rho +1)u\right]} =\frac{-1}{\rho+1}.$$