diff --git a/paddle/fluid/prim/api/composite_backward/composite_backward_api.h b/paddle/fluid/prim/api/composite_backward/composite_backward_api.h
index 48641d852537105dbbc4af98dae7410243ed1751..ffbeff8cade062b4065c60e02c4bdebc17f179d9 100644
--- a/paddle/fluid/prim/api/composite_backward/composite_backward_api.h
+++ b/paddle/fluid/prim/api/composite_backward/composite_backward_api.h
@@ -388,6 +388,14 @@ void expand_grad(const Tensor& x,
   }
 }
 
+template <typename T>
+void log_grad(const Tensor& x, const Tensor& out_grad, Tensor* x_grad) {
+  if (x_grad) {
+    // dx = dout / x
+    set_output<T>(out_grad / x, x_grad);
+  }
+}
+
 template <typename T>
 void exp_grad(const Tensor& out, const Tensor& out_grad, Tensor* x_grad) {
   if (x_grad) {
diff --git a/paddle/phi/api/yaml/backward.yaml b/paddle/phi/api/yaml/backward.yaml
index fde5deeafe9fb40d4ff26da79e777416d4d4449f..b70a752b2eca5bf55ec16dbd3595d9d0ca445679 100644
--- a/paddle/phi/api/yaml/backward.yaml
+++ b/paddle/phi/api/yaml/backward.yaml
@@ -804,6 +804,7 @@
   kernel :
     func : log_grad
   backward : log_double_grad
+  composite : log_grad(x, out_grad, x_grad)
   inplace : (out_grad -> x_grad)
 
 - backward_op : log_loss_grad
diff --git a/python/paddle/fluid/tests/unittests/test_activation_op.py b/python/paddle/fluid/tests/unittests/test_activation_op.py
index 591f4dc962c06fffba12fd8e4ed5260b760c4b1e..6883a769829cec7a66848cf521467fac41d84041 100755
--- a/python/paddle/fluid/tests/unittests/test_activation_op.py
+++ b/python/paddle/fluid/tests/unittests/test_activation_op.py
@@ -2623,10 +2623,15 @@ class TestLog(TestActivation):
     def setUp(self):
         self.op_type = "log"
         self.check_eager = True
+        self.prim_op_type = "prim"
         self.python_api = paddle.log
         self.init_dtype()
         self.init_shape()
 
+        if len(self.shape) == 0:
+            # for 0-D tensor, skip cinn testing
+            self.enable_cinn = False
+
         np.random.seed(1024)
         x = np.random.uniform(0.1, 1, self.shape).astype(self.dtype)
         out = np.log(x)
@@ -2637,7 +2642,7 @@ class TestLog(TestActivation):
     def test_check_grad(self):
         if self.dtype == np.float16:
             return
-        self.check_grad(['X'], 'Out', check_eager=True)
+        self.check_grad(['X'], 'Out', check_eager=True, check_prim=True)
 
     def test_error(self):
         in1 = paddle.static.data(name="in1", shape=[11, 17], dtype="int32")
@@ -3846,7 +3851,7 @@ create_test_act_fp16_class(TestSoftRelu, grad_atol=0.85)
 create_test_act_fp16_class(TestELU)
 create_test_act_fp16_class(TestCELU)
 create_test_act_fp16_class(TestReciprocal)
-create_test_act_fp16_class(TestLog)
+create_test_act_fp16_class(TestLog, check_prim=True)
 if core.is_compiled_with_rocm():
     create_test_act_fp16_class(TestLog2, atol=5e-2, grad_atol=0.85)
 else: