How to interpret the physical meaning of cointegration vectors of log prices in real world

I'm trying to understand the physical meaning of cointegration vectors of log prices in the real world. For example, if I have two assets $A$ and $B$ , and Johansen test gives us a cointegration coefficient of [1 -1] for their log prices: $$log(A)-log(B)=\eta$$ When we convert this relationship back to real prices to understand its physical meaning, we have $$\frac{A}{B}=e^\eta$$ This makes sense because a fixed ratio between two real assets is intuitive. However, for cointegration coefficient o