Three ideas here:
1) Don't use word
I am asuming you are using word as a midle step fo generate a pdf. So if that is the case don't use it.
Word is a realy bad option to handle raster images. It process the inserted images compressing them and resampling them.
Instead Save a copy of your work directly as pdf.
File > Save as > Choose pdf as the output format.
2) There is no need to export as a png
Depending on the efects on the original file the logo will stay as vector format. Try to avoid rasterizing it if it is not really necesary.
If it has some efects like gradients, transparencies or shadows, probably you will need to rasterize it, but the resolution and size depends on the aplication of the logo.
Search some info about resolution for printing. It can go from 300ppi to lets say 100ppi for a pdf which is going to be distributed on the internet.
3) In your specific workflow
There are a lot of things you can be doing... not right :o).
Lets think you really need to use word becouse the logo is part of a corporative letter or something like that.
There is a chance that the method you are using to convert from word to pdf is not configured properly. Search the options in your aplication and see if it has some options to change resolution of the raster images inside.
A pdf has too many factors to consider when exporting a file. It is not the same as just saving a file with another name. A pdf export process the file in several ways.
a) Resampling the raster images. Depending on the preset choosen.
b) Converting image mode. From rgb to grayscale or cmyk or leaving them as the original.
c) Flatening effects. LIke transparencies, but sometimes the exporter can make aditional objects flatening some zones.