 # Taking a logarithmic derivative of a function I have the following expression: log &#x2061;<! veirarer 2022-06-27 Answered
Taking a logarithmic derivative of a function
I have the following expression:
$\mathrm{log}\left(1-\frac{r}{{r}_{s}}\right)$
which I would like to take the following derivative of (and where ${r}_{s}$ is a constant):
$\frac{d\left(\mathrm{log}\left(1-\frac{r}{{r}_{s}}\right)\right)}{d\mathrm{log}\left(r\right)}$
What kind of strategies could I employ to find a solution?
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let $t=\mathrm{log}r$ then ${e}^{t}=r$ substituting in your expression you have to find
$\frac{d}{dt}\mathrm{log}\left(1-\frac{{e}^{t}}{{r}_{n}}\right)$
after differentiating retain the value of $r$

We have step-by-step solutions for your answer! Petrovcic2x
Consider that you have log(u). Then apply d(log(u))/du = 1 / u. Now, since you want, I guess, the derivative with respect to "r" and since u = 1 - r / rs, then du/dr = - 1 / rs and so,
d(log[1 - r / rs])/dr = d(Log(u))/du * du/dr = 1 / (r - rs).

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