I have
$$ \log\left(\frac{b\exp(a)}{1 - b\exp(a)}\right)$$
and I try to find the shortest representation. I found
$$ 0 - \log(\frac{1}{b\exp(a)} - 1)$$
and
$$ a - \log(b - \exp(a))$$
I'm wondering whether there is something better I could do. The clean setup gives me hope for more, but I fail to get anything else. I'm fine with approximations too; $a$ is negative and $b$ is positive. $a$ is potentially large, and $b \in (0, 1)$.
$a$ and $b$ are in no way related, be it in differences or products.
You have
$ - \log(\frac{1}{\exp(a)b} - 1)$
Let $a=-c$, with $c>0$ we get
$\log\left(\frac1{\frac{\exp(c)}{b} - 1} \right)=\log\left(\frac b{\exp(c) - b} \right)=\log(b)-\log(\exp(c)-b)$
For large $c$ the term is approximately $-c=a$.