Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 4 / c / i / Solution

An admissible change of Weierstrass equation of an elliptic curve replaces the $x$-coordinate by a degree-one rational function, so the two logarithmic heights differ by $O(1)$. Dividing this bounded difference by $4^r$ in the defining limit shows that the canonical height is unchanged. It is intrinsic to the elliptic curve and the chosen point.