Why is logistic regression based on an exponential function I don't entirely understand why logistic regression is based on an exponential function. 
The sigmoid function seems to assume that as the dependent variables increase, the independent variable starts to increase exponentially. How do we know this is true? How do we know that this isn't a linear relationship?
 A: [I didn't answer this before because I thought it would be a duplicate but I didn't locate a suitable one; I'll base a brief answer off my comments, at least until such a thread is located.]
Note that the model is for a probability $P(Y=1|\mathbf{X}=\mathbf{x})$
which must lie between 0 and 1. 
A linear function must eventually pass outside those limits, giving impossible probabilities. That's usually undesirable.
The logit is an example of a link function - the most popular - that stays within those limits. 
There are a variety of other functions that can be used for example, both probit and complementary-log-log links are sometimes used instead of the logit link for conditionally binomial models.
While convenient from several viewpoints, it's no more safe to assume it's specifically logistic than to assume it's anything else that satisfies the same restrictions, outside of the fact that it's often a reasonable approximation$-$as the famous saying goes, all models are wrong, but some are useful. 
It makes about as much sense as to assume a relationship is linear in ordinary regression. It's convenient and often a good approximation, but typically it would be unwise to think the relationships are actually exactly linear.
