I've stumbled upon the following approximation which I am a bit baffled by and I hope you could help me get some intuition about when it is reasonable and when it is not.
Let $c_i,a_i,b_i$ for $i=1,\dots,N$ be given and all positive.
It then reads $$ P(x) =C + \log (\; \sum_{i=1}^N c_i \exp(- ( a_i+ b_i x) \;) =C -\sum_{i=1}^N c_i ( a_i+ b_i x ) + \text{correction} $$ and further goes, since $P$ is then an approximately affine function it is well approximated by a first order Taylor expansion around $x^*$ (satisfying $P(x^*)=0$ known to exist).
Is it possible to say something about when this works well and not? Specifically the final Taylor approximation more than the intermediate step.