Shrink Image without Loss of Quality

[Ofnuts]

This is due to spatial frequency folding. The cure is to blur the image before you scale down, to remove the details that would be frequency-folded but wouldn't be visible in the scaled-down image anyway. Blur how much, you will ask... by the scaling ratio: 6900/500 in your case. You don't even need to compute this on the side, open Filter>Blur>Gaussian blue and enter 6900/500 in the Blur radius fields (should yield 13.8).

Unblurred:

   

Blurred:

   

This said, this logo likely exists as a vector image (.ai, .svg) and scaling it as a vector image (Inkscape, Illustrator) would yield perfect results.