Customize category URL

I’ve searched around and have not found the answers im looking for. By default I have:
– example.com/blog (blog archive)
– example.com/blog/category/announcements (single taxonomy view)

Now I’ve created a custom post type & taxonomy (categories) to go along with it, but I’m having trouble mirroring the blog’s URL structure. For example my goal is:
– example.com/talks (blog archive)
– example.com/talks/category/support (single taxonomy view)

Here is what I have so far:

function uc_add_talks_post_type() {
  $talks_labels = array(
    'name'                => 'Talks',
    'singular_name'       => 'Talk',
    'all_items'           => 'All Talks',
    'add_new'             => 'Add new',
    'add_new_item'        => 'Add new Talk',
    'edit_item'           => 'Edit Talk',
    'new_item'            => 'New Talk',
    'view_item'           => 'View Talk',
    'search_items'        => 'Search Talks',
    'not_found'           => 'No Talks found',
    'not_found_in_trash'  => 'No Talks found in Trash',
  );

  $talks_args = array(
    'labels'             => $talks_labels,
    'public'             => true,
    'publicly_queryable' => true,
    'show_ui'            => true,
    'menu_position'      => 5,
    'rewrite'            => array( 'slug' => 'talks', 'with_front' => false ),
    'has_archive'        => 'talks',
  );

  $talks_cat_args = array(
    'hierarchical' => true,
    'labels' => array(
      'name' => __( 'Talk categories', 'jointstheme' ),
      'singular_name' => __( 'Talk category', 'jointstheme' ),
      'search_items' =>  __( 'Search talk categories', 'jointstheme' ),
      'all_items' => __( 'All talk categories', 'jointstheme' ),
      'parent_item' => __( 'Parent talk category', 'jointstheme' ),
      'parent_item_colon' => __( 'Parent talk category:', 'jointstheme' ),
      'edit_item' => __( 'Edit talk category', 'jointstheme' ),
      'update_item' => __( 'Update talk category', 'jointstheme' ),
      'add_new_item' => __( 'Add new talk category', 'jointstheme' ),
      'new_item_name' => __( 'New talk category name', 'jointstheme' )
    ),
    'show_admin_column' => true,
    'show_ui' => true,
    'query_var' => true,
    'rewrite' => array(
      'slug' => 'talks',
      'with_front' => false
    ),
  );

  register_post_type( 'uc_talks', $talks_args );
  register_taxonomy( 'uc_talks_cat', array('uc_talks'), $talks_cat_args );
}

add_action( 'init', 'uc_add_talks_post_type');

(I tried to use the proper terminology here, taxonomy vs. category, archive vs. listing etc. But feel free to correct that if I messed up).