Erase from an object using a different object

Sorry, I'm a complete noon when it comes to Illustrator so please excuse my ignorance

I had a logo created for me, and now I'm having a problem where I can make a part of the image transparent because of the way it was originally created.

Basically, what was created was a swoosh, where a solid blue elliptical object was created and then a smaller solid white elliptical was placed on top of it.

Then text and other objects were placed on top of these.

What I'm trying to do is to make the white elliptical as a deletion or blank space of the larger blue elliptical to make that portion transparent.

This is an example of what I ended up with when exporting it and using an eraser tool in Paint:

What would the best way be to accomplish this?