Will Prism do a three-parameter or four-parameter logistic fit to my data?

The term "logisitic" is used in many ways.  Read about the various uses of the term logistic.. While Prism cannot do logistic regression, it can use nonlinear regression to fit log(dose) vs. response curves. The models used to fit these data are sometimes called logistic models. 

If you use Prism 5 or 6:

From your data table where X is log(concentration) and Y is response, click Analyze and choose nonlinear regression. Then choose the  "Dose-response - Stimulation" or "Dose-response - Inhibition" families of equations. The equations with 'variable slope' have four parameters (top plateau, bottom plateau, middle or logEC50, and slope) so are sometimes called 4PL (four parameter logistic). The first equation in each group has three parameers (top, bottom, and middle, assuming standard slope) so is sometimes called 3PL.  Prism can also fit an asymmetric, five parameter, "logistic" equation (5PL). 

Prism can guide you. Create an XY table, and choose  one of the dose-response sample data sets. Along with the sample data, Prism provides instructions. 

The built in equation assumes X is log(concentration). If you prefer X to be concentration itself, see this page. 
If you use Prism 4:
 For the three-parameter fit, choose the "Sigmoidal dose-response" model from the "Classic equations" group. For the four-parameter fit, choose the "Sigmoidal dose-response (variable slope)" model.

Step by step examples: RIA/ELISA      Dose response curve.

If you use Prism 3:
Step by step example for Prism 3.