I'm considering a parametrized sigmoid function such as the following logistic:
$$f(x)=\frac{e^{a+bx}}{1+e^{a+bx}}$$
And I'm interested only in the interval $\displaystyle x >= 0$ and $\displaystyle x < x_{max}$ (with a given $x_{max}$).
Two tiny questions regarding its properties:
1) Is the sum of sigmoid functions always a sigmoid function itself?
2) Is the log function (i.e. $\displaystyle f(x) = log_{a}(bx^{c})$ or similar) a particular case of a sigmoid function? In other words, can I pick parameters for the logistic function above so that it behaves like a log function in the given interval?
Thanks.