a relationship based on love and trust will give you the best foundation on which to raise a contented dog. "on which"用法？

不定詞（片語）當形容詞用時，被修飾的名詞與不定詞的邏輯關係可能是下列三類之一：
1. I'm looking for someone to help me with my work.不定詞的主詞是someone
2. I have a lot of homework to do.不定詞的受詞是homework
3. I don't have enough money to buy those shoes (with).被修飾的名詞money既不是不定詞的主詞也不是受詞。不定詞與該名詞的關係建立在一個可以省略，但不至誤解句意的介系詞上(with xxx)，是該介系詞的「受詞」。to buy those shoes with the money。

你的問題屬於第三類: ... to raise a contented dog on that foundation.
on which建立了不定詞to raise與被其修飾的名詞間的關係。關係代名詞which = that foundation，當介系詞on的受詞。其實這裡的on which是可以省略的，不致影響句意。

how=by which;
eg:-A relationship based on love and trust will give you the best foundation (by which) to raise a contented dog.

A relationship based on love and trust will give you the best foundation (on which) to raise a contented dog.
= A relationship based on love and trust will give you the best foundation (where it is the best relationship) to raise a contented dog.

on which = where，which 指 foundation，原句中 on which 也可省去
on which 或 where 後應接子句，但 it is the best relationship 和前述 the best foundation 語意和用字重覆，為求精簡，省去。

>那什麼時候需要省去什麼時候留下比較好？
寫作沒有固定的法則，純粹是個人的取捨，只要語意清楚不囉嗦，讓人讀起來通順流暢，不讓人厭煩就行。

on which 後的寫法不只一種，
...on which (a reliable environment can be created) to raise a contented dog.
...on which (the owner can nurture the intimacy) to raise a contented dog.
作者選擇省去，也是個人的取捨。

X which的用法要很句子的內容的preposition主要的意義。

這句的用詞是best foundation，因此foundation的preposition必定是"on"，建在這基礎"上"，所以連後句的which所指是這foundation"上"，所以用on which.

當另一句的另一說法或用詞是另一種presposition的話，X which便會改為配合的preposition.

例如：The report analyzed several causes of the accident in which 45 people died.
這句which前所指的是the accident這報告所要調查分析，後連句指出"在"這意外"中"死了四十五個人，因此which的preposition便使用配合"in the accident"的"in"。

The report of this accident is very poorly written of which the cause of the accident is analyzed.
這句的which是指這report的寫作/內容，因此配合這report是關乎"於"這意外，因此後連句便配合report of this accident的習慣presposition"關於"一樣東西，of which意思類同詳解一下"關於"這報告。

當然不是每種句式都會"顯示"前句或要寫的X which的X是甚麼，你要識用便要多看書和理解當一句句子連續"解釋"或"詳細"或"深入"講多一兩個重點時，那主句的主題或內容上所慣用的preposition是甚麼，才能看看那一種是配合"同義"的。which一般會是in which , on which, of which, at which, of which.