diff --git a/moto/logs/models.py b/moto/logs/models.py
index 9025e8d82..98718c6a3 100644
--- a/moto/logs/models.py
+++ b/moto/logs/models.py
@@ -333,7 +333,7 @@ class LogGroup(BaseModel):
         self, log_group_name, log_stream_name, log_events, sequence_token
     ):
         if log_stream_name not in self.streams:
-            raise ResourceNotFoundException()
+            raise ResourceNotFoundException("The specified log stream does not exist.")
         stream = self.streams[log_stream_name]
         return stream.put_log_events(
             log_group_name, log_stream_name, log_events, sequence_token
diff --git a/tests/test_logs/test_logs.py b/tests/test_logs/test_logs.py
index 79dcb2d3e..4e4b222de 100644
--- a/tests/test_logs/test_logs.py
+++ b/tests/test_logs/test_logs.py
@@ -46,12 +46,15 @@ def test_exceptions():
         logEvents=[{"timestamp": 0, "message": "line"}],
     )
 
-    with pytest.raises(ClientError):
+    with pytest.raises(ClientError) as ex:
         conn.put_log_events(
             logGroupName=log_group_name,
             logStreamName="invalid-stream",
             logEvents=[{"timestamp": 0, "message": "line"}],
         )
+    error = ex.value.response["Error"]
+    error["Code"].should.equal("ResourceNotFoundException")
+    error["Message"].should.equal("The specified log stream does not exist.")
 
 
 @mock_logs