On the right you can see the plain scene for the logo. The logo is standing on a white plane so the background of the image is white and the logo is casting a shadow on the plane. See the source code of this scene for more details.

To render the scene with a transparent background you have to add the no_image flag to the plane object. This tells POV-Ray to not consider the object for direct camera rays so it does not show up directly in the image. Rendering with +UA then makes the background of the image transparent like visible on the right. This however also removes the shadow the logo is casting on the plane.

To get both the shadow and the transparent background you have to do multiple renders and combine them. In the scene we therefore define 3 constants for the three renders to create:

#declare VARIANT_ALPHA=1;   // alpha render (for foreground)
#declare VARIANT_BKG=2;     // background render (for shadow color)
#declare VARIANT_SHADOW=3;  // shadow render (for shadow shape)

and set up a symbol Variant to define the current render variant.

The first render needed is the one above for the logo itself. Then we have to render the scene with only the shadow. For this the no_image flag is applied to the logo itself while the plane is kept visible. The plane finish can be adjusted for the intended shadow color.

In addition we render a shadow mask to steer the transparency of the shadow. The shadow itself should be partly transparent while the rest of the background is completely transparent. To get such an image the we modify the previous setup with a different finish for the plane and turn off radiosity.

The resulting background plane definition looks like

plane {
  z, 0.0
  texture { 
    pigment { color rgb 1 }
    #switch (Variant)
      #case (VARIANT_ALPHA)
        finish { ambient 0 diffuse 0.5 }
      #break
      #case (VARIANT_BKG)
        finish { ambient 0 diffuse 0.25 }
      #break
      #case (VARIANT_SHADOW)
        finish { ambient 0.05 diffuse 1.05 }
      #break
    #end
  }
  #if (Variant = VARIANT_ALPHA)
    no_image
  #end
}

See the complete scene for the other differences.

The three renders can be combined to a single image with a image editing program of your choice. The third variant should be added to the second(background) variant as alpha channel and the first(foreground) variant is copied on top of that. Using the ImageMagick tools this task and the three renders can be issued with a simple shell script. The resulting image can be found in large on the right and with various background colors below.