Thanks for the edit. Much Clearer (and easier to accomplish) than it appeared initially.
All you need is an ellipses with a soft edge. That can be accomplished a couple ways:
1.) The Feather Effect
Draw the ellipse and merely choose
Effect > Stylize > Feather and enter a value. The higher the value the more the edge will blur. This blends inward from the edge of the ellipse.
2). A Gradient
Draw the ellipse and fill it with a black to black radial gradient. Yes, set both color stops to the same color. However, set one color stop's opacity to 0%.
Switch to the Gradient Tool and using the on screen Gradient Annotator you can alter the gradient to match the shape of the ellipse, control where the drop off occurs, etc. This method gives more "live" control.
For an actual logo using soft edges, such as this, is generally a bad idea. But I don't know the actual implementation you may be envisioning.