In color image processing, the objective of image completion is to restore missing entries from the incomplete observation image. Recent improvements have assisted in resolving the rank minimization issue, thereby promoting the realization of completion. This paper adopts the new quaternion matrix logarithmic norm to approximate rank in accordance with the quaternion matrix framework. Unlike the traditional matrix completion method, which handles RGB channels separately, the quaternion-based method avoids the destruction of the structure of images by placing the color image in a pure quaternion matrix. Furthermore, the logarithmic norm induces a more accurate rank surrogate. Based on the logarithmic norm, it is possible to exploit not only the factorization strategy but also the truncated technique, thus achieving successful image restoration. The alternating minimization framework renders it possible to optimize the two strategies, and mathematically detailed validation of the convergence analysis is provided. The experimental results demonstrate that the use of logarithmic surrogates in the quaternion domain is a superior strategy for solving the problem of color image completion.