Can I approximate log(1-x) with -x, if my ?
If i have an optimization problem: maximize A*x + B*log(1-x)
can I exchange with a problem: maximize A*x + B*(-x) ?
Prove it has given you the series of expansion. For increasing powers of x will get smaller so you can simply say high powers are roughly 0 compared to low powers and discard the former.
At which value of x this applies depends on how accurate your function needs to be.