Properties of sigmoid functions 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.

I am assuming that a sigmoid function is bounded, and its graph has exactly one inflection point (among other properties).
For (1), then, the answer is no.  If $f(x)=\frac{e^{3x}}{1+e^{3x}}$ and $g(x)=\frac{e^{2x-5}}{1+e^{2x-5}}$, then $h(x)=f(x)+g(x)$ can be seen to have two points of inflection, so h(x) is not sigmoid. 
For (2), your function is unbounded on $0\le x \le x_{\mbox{max}}$, but sigmoid functions are bounded.  So these functions have decidedly different behavior.  Though, it does depend somewhat on what you mean by "like".
