Match taxonomy paths to former blog pattern

@ju52

My ask is less about segmenting tagged content and more about preserving the URL from my existing structure. For example if I used different tag name for my blog post than for my page posts I would end up with the following.

http://example.com/blog/
http://example.com/pages
http://example.com/ctags
http://example.com/ptags

When what I want is

http://example.com/blog/tag/

Taxonomies are global, and their URL to their index page is rooted at the site root. I want a taxonomy that is rooted at /blog

Edit: to clarify I am also looking for individual tags, not just an index page for all tags:

http://example.com/blog/tag
http://example.com/blog/tag/cats
http://example.com/blog/tag/dogs