InDesign: Drop Shadow Knocking Out of Background

As roaringmousegraphics suggests it sounds like a blending mode error is causing this. It's unclear to me whether you have the shadow applied in the Illustrator file itself or if you're applying the drop shadow after importing the logo into InDesign. The screen shots below show where to check the blending mode for your drop shadow both in Illustrator (on the left) or InDesign (on the right). To check whether a previously applied drop shadow has the correct blending mode in Illustrator select the object with the drop shadow and then open the Appearance Panel and double-click on the entry found there for Drop Shadow. To check whether a previously applied drop shadow has the correct blending mode in InDesign select the object with the drop shadow, open the Effects Panel and double-click on the entry there for "Object:". In both cases the correct mode for a drop shadow is "Multiply".