When your navbar gets narrower with scrolling, your logo will shrink. Here are some scripts with a demo on Fiddle or Codepen that you can use for this purpose.

Red square

This red square shrinks or grows with scrolling. The square is made with a div. [Demo]

Stackoverflow logo

A logo with image shrinks with scrolling. [Demo]

Growing bottle image

A bottle grows from invisible to big by scrolling. [Demo]

Shrinking image

Image shrinking on scroll. [Demo]

Shrinking image

Shrinking topbar. [Demo]