The excerpt I took from one book, is telling me that sigmoid different name is logit.
What is the reasoning behind? I don't get it.
I know the sigmoid function is defined:
$sigmoid(x) = \large \frac{exp(x)}{1+exp(x)} $
I know the logit function is defined as:
$logit(x) = \large \log \frac{x}{1-x}$
And I even know these two functions are inverse.
The logistic function has its name from the property that it starts off exponentially (like unrestricted population growth) and later grows slower until it asymptotically stops (as if due to resource exhaustion, i.e., "logistic" problems with "feeding" the population). The function $x\mapsto\frac{e^x}{1+e^x}$ matches this description well because for $x\ll 0$, this is $\approx e^x$ and for $x\gg0$ this is $\approx 1$. As this is also one of many functions whose graph is S-shaped, it is also called sigmoid.
However, logit is usually defined as $x\mapsto \ln \frac x{1-x}$.