ABSTRACT Low‐rank tensor completion (LRTC) aims to recover a complete low‐rank tensor from incomplete observed tensor, attracting extensive attention in various practical applications such as image processing and computer vision. However, the current methods often perform well only when there is a sufficient of observed information, and they perform poorly or may fail when the observed information is less than 5%. To improve the utilization of observed information, a new method called the tensor joint rank with logarithmic composite norm (TJLC) method is proposed. This method simultaneously exploits two types of tensor low‐rank structures, namely tensor Tucker rank and tubal rank, thereby enhancing the inherent correlations between known and missing elements. To address the challenge of applying two tensor ranks with significantly different directly to LRTC, a new tensor logarithmic composite norm is further proposed. Subsequently, the TJLC model and algorithm for the LRTC problem are proposed. Additionally, theoretical convergence guarantees for the TJLC method are provided. The experiments on various real datasets demonstrate that the proposed method outperforms state‐of‐the‐art methods significantly. Particularly, the proposed method achieves satisfactory recovery even when the observed information is as low as 1%, and the recovery performance improves significantly as the observed information increases.