The　low-rank　tensor　could　characterize　inner　structure　and　explore　high-order　correlation　among　multi-view　representations,　which　has　been　widely　used　in　multi-view　clustering.　Existing　approaches　adopt　the　tensor　nuclear　norm　(TNN)　as　a　convex　approximation　of　non-convex　tensor　rank　function.　However,　TNN　treats　the　different　singular　values　equally　and　over-penalizes　the　main　rank　components,　leading　to　sub-optimal　tensor　representation.　In　this　paper,　we　devise　a　better　surrogate　of　tensor　rank,　namely　the　tensor　logarithmic　Schatten-p　norm　(TLSpN),　which　fully　considers　the　physical　difference　between　singular　values　by　the　non-convex　and　non-linear　penalty　function.　Further,　a　tensor　logarithmic　Schatten-p　norm　minimization　(TLSpNM)-based　multi-view　subspace　clustering　(TLSpNM-MSC)　model　is　proposed.　Specially,　the　proposed　TLSpNM　can　not　only　protect　the　larger　singular　values　encoded　with　useful　structural　information,　but　also　remove　the　smaller　ones　encoded　with　redundant　information.　Thus,　the　learned　tensor　representation　with　compact　low-rank　structure　will　well　explore　the　complementary　information　and　accurately　characterize　the　high-order　correlation　among　multi-views.　The　alternating　direction　method　of　multipliers　(ADMM)　is　used　to　solve　the　non-convex　multi-block　TLSpNM-MSC　model　where　the　challenging　TLSpNM　problem　is　carefully　handled.　Importantly,　the　algorithm　convergence　analysis　is　mathematically　established　by　showing　that　the　sequence　generated　by　the　algorithm　is　of　Cauchy　and　converges　to　a　Karush-Kuhn-Tucker　(KKT)　point.　Experimental　results　on　nine　benchmark　databases　reveal　the　superiority　of　the　TLSpNM-MSC　model.