Detecting targets in dynamically changing clutter environments has always been challenging for radar techniques. The innovative matrix information geometry-based (MIG) detector, which equips with different geometric distances to discriminate the target and clutter on the Hermitian positive-definite manifold, treats the detection issue from a geometric point of view, however, the current geometric distances do not always perform satisfactory under the changing environment. To address this problem, this article proposes an adaptive log-Euclidean (LE) metric on the manifold to properly match the actual clutter environment and achieve satisfactory performance. Specifically, the adaptive LE metric is developed on a dimension-reduced manifold by deriving a discriminative manifold transformation, and the determination of the transformation is formulated as a joint optimization problem between the cell under test and the clutter centroid. In this study, manifold-based methods and an eigen-decomposition-based method are proposed to solve the optimization problem. Moreover, an adaptive LE metric-based MIG detector is designed, and its bounded constant false alarm rate property is analyzed. At last, both simulated data and real radar data are utilized to validate the performance of the proposed method. Particularly, in the unmanned aerial vehicle target detection experiment, the results show that the strong clutter is effectively suppressed and the target is simultaneously highlighted without any prior knowledge.