When I switch from an ordinary (linear) X axis to a log axis, the curves become incomplete or ragged

Prism (versions 1-4) define a curve a series of line segments, equally spaced on a linear X axis. When you convert to a log axis, these segments are not equally spaced, and you see gaps at the beginning, and line segments rather than curves.. The more log units your data span, the more pronounced the problem.

The best fix is to upgrade to Prism 5. When you change your axis to log, Prism 5 simply changes how it plots the curves. It just works. You don't have to do anything special.

For earlier Prism versions, there are two work-arounds.

The simple solution is to tell Prism to make many more line segments. You set this in the "Output..." options subdialog under "Parameters: Nonlinear regression". With more line segments, it matters less when they get stretched to a log axis. Whether or not this will work for you depends on how many log units your data cover

A better (but more difficult)  solution is to transform the data to logs, instead of using a logarithmic axis on the graph. Do the transform X=log (X). Now the graph has a linear scale, but you can use the "Power of ten"  or "Antilog" tick labeling format to show the scale. Then rewrite your equation, changing every quantity "X" to "10^X". That's because X is now log of concentration, so you need to take the antilog to make it fit. This is the approach Prism's built-in sigmoidal equations use. Follow a step-by-step tutorial.