What to do With Curvilinear Relationships? If I have determined a curvilinear relationship between my dichotomous y and continuous x, what should I do before running a logit regression? 
Should I log transform my x variable, etc?
 A: There is not enough information here for a definitive answer, but two possibilities are:
1) Adding polynomial terms of X (e.g. $x^2, x^3$)
2) Using some sort of spline curve (e.g. restricted cubic splines) of x.
The choice would depend on a few things:
1) How complex is the curve? Very complex relationships are better served by splines.
2) How "explicable" does the model have to be? It is generally easier for people to understand polynomial terms (at least, if you don't go beyond $x^3$) than splines.
I would use a log transform only if the log transform makes substantive sense. Such a transform changes additive operations to multiplicative ones. That is, if the IV is now log(x), then it implies that (say) doubling x will have a uniform effect on y, rather than (say) adding 10 to x having a uniform effect on y. Log transforms often make sense for amounts, in particular monetary amounts. For example, it seems reasonable that doubling income ought to have a constant effect. 
One case where splines have been well used (as an example) is the relationship between age and height over the human lifespan: Height rises quickly, then slows, then speeds up, then gradually slows to a plateau and then (much later) declines slowly. That would be very hard to model with a polynomial!
A: One must be a bit careful in curve-fitting. One of my professors once commented during a presentation (by someone else) that had a high-order curve fit: "With five free parameters, one can curve-fit an elephant. With eight, one can fit a running elephant."
Somewhere between a least squares line (or the L1 or L_inf equivalent) and a full Lagrange interpolation polynomial should lie a useful curve. Sometimes the form or even some of the parameters in a curve fit will be obvious from the conditions of the problem. This decision ends up being what I like to call a "physics" or "engineering" or "real world" decision, not so much a mathematical decision. How well a curve fits the data can be given a mathematical description; the meaning of such a fit cannot.
A method I have used in such cases is to separate the data into two parts; use one part to fit a curve; then use the left over so see how good the fit is. Another way that may be better is to bootstrap the system. With N data points (assuming just an X and a Y for each point), one draws (with replacement) N points from the data set and the computes the proposed fit. This is done some number of times. When a sample of fits is obtained, the variance of this sample is an indication of how well the fit fits the data. (fit fits?) A smaller variance can indicate that one type of fit is better than another but not whether such a fit is meaningful.
