diff --git a/paddle/fluid/operators/activation_op.cc b/paddle/fluid/operators/activation_op.cc
index ce53209bc3886b6b042e93732eba88c6c0984d8c..060a190f38e8acd42656ed40c5168116ed852263 100644
--- a/paddle/fluid/operators/activation_op.cc
+++ b/paddle/fluid/operators/activation_op.cc
@@ -839,6 +839,28 @@ class SquareDoubleGradMaker : public ::paddle::framework::SingleGradOpMaker<T> {
   }
 };
 
+// log Grad: dx = dout / x
+// log Grad Grad: ddout = ddx / x; dx = -(dout / x) * (ddx / x)
+template <typename T>
+class LogDoubleGradMaker : public ::paddle::framework::SingleGradOpMaker<T> {
+ public:
+  using ::paddle::framework::SingleGradOpMaker<T>::SingleGradOpMaker;
+
+ protected:
+  void Apply(GradOpPtr<T> op) const override {
+    op->SetType("log_grad_grad");
+    op->SetInput("X", this->Input("X"));
+    // X@GRAD@GRAD: ddx
+    op->SetInput("DDX", this->OutputGrad(framework::GradVarName("X")));
+    op->SetInput("DOut", this->Input(framework::GradVarName("Out")));
+    op->SetAttrMap(this->Attrs());
+    // X@GRAD: dx
+    op->SetOutput("DX", this->InputGrad("X"));
+    // Out@GRAD@GRAD: ddy
+    op->SetOutput("DDOut", this->InputGrad(framework::GradVarName("Out")));
+  }
+};
+
 DECLARE_INPLACE_OP_INFERER(ActivationGradOpInplaceInference,
                            {framework::GradVarName("Out"),
                             framework::GradVarName("X")});
@@ -1219,3 +1241,32 @@ REGISTER_OP_CPU_KERNEL(
     ops::ActivationDoubleGradKernel<plat::CPUDeviceContext,
                                     ops::AbsGradGradFunctor<int64_t>>);
 /* ========================================================================== */
+
+/* ==========================  Log register ==================================*/
+REGISTER_OPERATOR(
+    log, ops::ActivationOp, ops::LogOpMaker, ops::ActivationOpInferVarType,
+    ops::ActivationGradOpMaker<ops::LogGradFunctor<float>::FwdDeps(),
+                               paddle::framework::OpDesc>,
+    ops::ActivationGradOpMaker<ops::LogGradFunctor<float>::FwdDeps(),
+                               paddle::imperative::OpBase>,
+    ops::ActFwdInplaceInferer);
+REGISTER_OPERATOR(log_grad, ops::ActivationOpGrad,
+                  ops::ActivationGradOpInplaceInference,
+                  ops::LogDoubleGradMaker<paddle::framework::OpDesc>,
+                  ops::LogDoubleGradMaker<paddle::imperative::OpBase>);
+
+REGISTER_OPERATOR(
+    log_grad_grad,
+    ops::ActivationOpDoubleGrad<ops::LogGradGradFunctor<float>::FwdDeps()>,
+    ops::ActivationDoubleGradOpInplaceInference);
+
+REGISTER_ACTIVATION_CPU_KERNEL(log, Log, LogFunctor, LogGradFunctor);
+
+REGISTER_OP_CPU_KERNEL(
+    log_grad_grad, ops::LogDoubleGradKernel<plat::CPUDeviceContext,
+                                            ops::LogGradGradFunctor<float>>,
+    ops::LogDoubleGradKernel<plat::CPUDeviceContext,
+                             ops::LogGradGradFunctor<double>>,
+    ops::LogDoubleGradKernel<plat::CPUDeviceContext,
+                             ops::LogGradGradFunctor<plat::float16>>);
+/* ========================================================================== */
diff --git a/paddle/fluid/operators/activation_op.cu b/paddle/fluid/operators/activation_op.cu
index 072d952d2618d2c9dbbe27641dcd951a634753a0..839776ad58d0352cd9bdd59530951e4eea1120b3 100644
--- a/paddle/fluid/operators/activation_op.cu
+++ b/paddle/fluid/operators/activation_op.cu
@@ -193,3 +193,15 @@ REGISTER_OP_CUDA_KERNEL(
     ops::ActivationDoubleGradKernel<paddle::platform::CUDADeviceContext,
                                     ops::AbsGradGradFunctor<int64_t>>);
 /* ========================================================================== */
+
+/* ==========================  Log register ==================================*/
+REGISTER_ACTIVATION_CUDA_KERNEL(log, Log, LogFunctor, LogGradFunctor);
+
+REGISTER_OP_CUDA_KERNEL(
+    log_grad_grad, ops::LogDoubleGradKernel<plat::CUDADeviceContext,
+                                            ops::LogGradGradFunctor<float>>,
+    ops::LogDoubleGradKernel<plat::CUDADeviceContext,
+                             ops::LogGradGradFunctor<double>>,
+    ops::LogDoubleGradKernel<plat::CUDADeviceContext,
+                             ops::LogGradGradFunctor<plat::float16>>);
+/* ========================================================================== */
diff --git a/paddle/fluid/operators/activation_op.h b/paddle/fluid/operators/activation_op.h
index 3818a37e93ed183e29105e801290b3047ce4ce11..507c5750e3b0abbb50913a59543bf3137f944c9f 100644
--- a/paddle/fluid/operators/activation_op.h
+++ b/paddle/fluid/operators/activation_op.h
@@ -1551,6 +1551,10 @@ class SquareDoubleGradKernel
   }
 };
 
+template <typename DeviceContext, typename Functor>
+class LogDoubleGradKernel
+    : public SquareDoubleGradKernel<DeviceContext, Functor> {};
+
 template <typename DeviceContext, typename Functor>
 class ELUDoubleGradKernel
     : public framework::OpKernel<typename Functor::ELEMENT_TYPE> {
@@ -1740,6 +1744,37 @@ class PowGradKernel
     functor(*place, x, out, dout, dx);
   }
 };
+
+template <typename T>
+struct LogGradGradFunctor : public BaseActivationFunctor<T> {
+  template <typename Device>
+  void operator()(const Device& dev, const framework::Tensor* X,
+                  const framework::Tensor* ddX, framework::Tensor* ddOut,
+                  const framework::Tensor* dOut, framework::Tensor* dX) const {
+    auto* d = dev.eigen_device();
+    auto ddx = framework::EigenVector<T>::Flatten(
+        GET_DATA_SAFELY(ddX, "Input", "DDX", "LogGradGrad"));
+    auto x = framework::EigenVector<T>::Flatten(
+        GET_DATA_SAFELY(X, "Input", "X", "LogGradGrad"));
+    // ddout = ddx / x; dx = -(dout / x) * (ddx / x)
+    // calculate dx first, so ddout can inplace ddx
+    if (dX) {
+      auto dout = framework::EigenVector<T>::Flatten(
+          GET_DATA_SAFELY(dOut, "Output", "DOut", "LogGradGrad"));
+      auto dx = framework::EigenVector<T>::Flatten(
+          GET_DATA_SAFELY(dX, "Output", "DX", "LogGradGrad"));
+      dx.device(*d) = dout * static_cast<T>(-1) * ddx / (x * x);
+    }
+    if (ddOut) {
+      auto ddout = framework::EigenVector<T>::Flatten(
+          GET_DATA_SAFELY(ddOut, "Output", "DDOut", "LogGradGrad"));
+      ddout.device(*d) = ddx * static_cast<T>(1) / x;
+    }
+  }
+
+  static constexpr ActBwdOpFwdDeps FwdDeps() { return kDepX; }
+};
+
 }  // namespace operators
 }  // namespace paddle
 
@@ -1758,7 +1793,6 @@ class PowGradKernel
   __macro(asin, Asin, AsinFunctor, AsinGradFunctor);                          \
   __macro(round, Round, RoundFunctor, ZeroGradFunctor);                       \
   __macro(reciprocal, Reciprocal, ReciprocalFunctor, ReciprocalGradFunctor);  \
-  __macro(log, Log, LogFunctor, LogGradFunctor);                              \
   __macro(log1p, Log1p, Log1pFunctor, Log1pGradFunctor);                      \
   __macro(brelu, BRelu, BReluFunctor, BReluGradFunctor);                      \
   __macro(soft_relu, SoftRelu, SoftReluFunctor, SoftReluGradFunctor);         \
diff --git a/python/paddle/fluid/tests/unittests/test_activation_nn_grad.py b/python/paddle/fluid/tests/unittests/test_activation_nn_grad.py
index e8b8a45fb677568947be82a1c77e6f2e7a17cdc1..bd51c3acc294f6cfa3ac9a41e16035c27d4b82eb 100644
--- a/python/paddle/fluid/tests/unittests/test_activation_nn_grad.py
+++ b/python/paddle/fluid/tests/unittests/test_activation_nn_grad.py
@@ -123,6 +123,30 @@ class TestSqrtDoubleGradCheck(unittest.TestCase):
             self.func(p)
 
 
+class TestLogDoubleGradCheck(unittest.TestCase):
+    @prog_scope()
+    def func(self, place):
+        shape = [2, 3, 7, 9]
+        eps = 1e-6
+        dtype = np.float64
+
+        x = layers.data('x', shape, False, dtype)
+        x.persistable = True
+        y = layers.log(x)
+
+        x_arr = np.random.uniform(0.1, 1, shape).astype(dtype)
+
+        gradient_checker.double_grad_check(
+            [x], y, x_init=x_arr, place=place, eps=eps)
+
+    def test_grad(self):
+        places = [fluid.CPUPlace()]
+        if core.is_compiled_with_cuda():
+            places.append(fluid.CUDAPlace(0))
+        for p in places:
+            self.func(p)
+
+
 class TestSquareDoubleGradCheck(unittest.TestCase):
     @prog_scope()
     def func(self, place):