diff --git a/moto/awslambda/responses.py b/moto/awslambda/responses.py
index b00dc7138..c06eb7b69 100644
--- a/moto/awslambda/responses.py
+++ b/moto/awslambda/responses.py
@@ -269,7 +269,10 @@ class LambdaResponse(BaseResponse):
             elif request.headers.get("X-Amz-Invocation-Type") == "DryRun":
                 status_code = 204
             else:
-                if request.headers.get("X-Amz-Log-Type") != "Tail":
+                if (
+                    request.headers.get("X-Amz-Log-Type") != "Tail"
+                    and "x-amz-log-result" in response_headers
+                ):
                     del response_headers["x-amz-log-result"]
                 status_code = 200
             return status_code, response_headers, payload
diff --git a/moto/awslambda_simple/models.py b/moto/awslambda_simple/models.py
index b1446b13e..68b7e32aa 100644
--- a/moto/awslambda_simple/models.py
+++ b/moto/awslambda_simple/models.py
@@ -51,6 +51,8 @@ class LambdaSimpleBackend(BaseBackend):
         response_headers: Any,
     ) -> Optional[Union[str, bytes]]:
 
+        if body:
+            return str.encode(body)
         return b"Simple Lambda happy path OK"
 
 
diff --git a/tests/test_awslambda_simple/test_lambda_simple.py b/tests/test_awslambda_simple/test_lambda_simple.py
index 3d0dccfd2..e316b7890 100644
--- a/tests/test_awslambda_simple/test_lambda_simple.py
+++ b/tests/test_awslambda_simple/test_lambda_simple.py
@@ -1,3 +1,4 @@
+import json
 from unittest import SkipTest
 import boto3
 
@@ -15,6 +16,36 @@ if settings.TEST_SERVER_MODE:
 @mock_iam
 @mock_lambda_simple
 def test_run_function():
+    # Setup
+    client = setup_lambda()
+
+    # Execute
+    result = client.invoke(
+        FunctionName=FUNCTION_NAME,
+        LogType="Tail",
+    )
+
+    # Verify
+    assert result["StatusCode"] == 200
+    assert result["Payload"].read().decode("utf-8") == "Simple Lambda happy path OK"
+
+
+@mock_iam
+@mock_lambda_simple
+def test_run_function_no_log():
+    # Setup
+    client = setup_lambda()
+    payload = {"results": "results"}
+
+    # Execute
+    result = client.invoke(FunctionName=FUNCTION_NAME, Payload=json.dumps(payload))
+
+    # Verify
+    assert result["StatusCode"] == 200
+    assert json.loads(result["Payload"].read().decode("utf-8")) == payload
+
+
+def setup_lambda():
     client = boto3.client("lambda", LAMBDA_REGION)
     zip_content = get_test_zip_file1()
     function_name = FUNCTION_NAME
@@ -30,10 +61,4 @@ def test_run_function():
         MemorySize=128,
         Publish=True,
     )
-
-    result = client.invoke(
-        FunctionName=FUNCTION_NAME,
-        LogType="Tail",
-    )
-    assert result["StatusCode"] == 200
-    assert result["Payload"].read().decode("utf-8") == "Simple Lambda happy path OK"
+    return client