diff --git a/moto/batch/models.py b/moto/batch/models.py
index 7674480c3..01c4a71fd 100644
--- a/moto/batch/models.py
+++ b/moto/batch/models.py
@@ -473,6 +473,8 @@ class Job(threading.Thread, BaseModel, DockerModel, ManagedState):
         self.name = "MOTO-BATCH-" + self.job_id
 
         self._log_backend = log_backend
+        self._log_group = "/aws/batch/job"
+        self._stream_name = f"{self.job_definition.name}/default/{self.job_id}"
         self.log_stream_name: Optional[str] = None
 
         self.attempts: List[Dict[str, Any]] = []
@@ -729,12 +731,12 @@ class Job(threading.Thread, BaseModel, DockerModel, ManagedState):
                 logs = sorted(logs, key=lambda log: log["timestamp"])
 
                 # Send to cloudwatch
-                log_group = "/aws/batch/job"
-                stream_name = f"{self.job_definition.name}/default/{self.job_id}"
-                self.log_stream_name = stream_name
-                self._log_backend.ensure_log_group(log_group, None)
-                self._log_backend.create_log_stream(log_group, stream_name)
-                self._log_backend.put_log_events(log_group, stream_name, logs)
+                self.log_stream_name = self._stream_name
+                self._log_backend.ensure_log_group(self._log_group, None)
+                self._log_backend.create_log_stream(self._log_group, self._stream_name)
+                self._log_backend.put_log_events(
+                    self._log_group, self._stream_name, logs
+                )
 
                 result = container.wait() or {}
                 exit_code = result.get("StatusCode", 0)
diff --git a/moto/batch_simple/models.py b/moto/batch_simple/models.py
index 243c5ce8e..2f985e058 100644
--- a/moto/batch_simple/models.py
+++ b/moto/batch_simple/models.py
@@ -79,6 +79,7 @@ class BatchSimpleBackend(BaseBackend):
 
         # We don't want to actually run the job - just mark it as succeeded
         job.job_started_at = datetime.datetime.now()
+        job.log_stream_name = job._stream_name
         job._start_attempt()
         job._mark_stopped(success=True)
 
diff --git a/tests/test_batch/test_batch_jobs.py b/tests/test_batch/test_batch_jobs.py
index 48f7a4eb9..05d93898f 100644
--- a/tests/test_batch/test_batch_jobs.py
+++ b/tests/test_batch/test_batch_jobs.py
@@ -234,6 +234,7 @@ def test_terminate_job():
     resp["jobs"][0]["jobName"].should.equal("test1")
     resp["jobs"][0]["status"].should.equal("FAILED")
     resp["jobs"][0]["statusReason"].should.equal("test_terminate")
+    resp["jobs"][0]["container"].should.have.key("logStreamName")
 
     ls_name = f"{job_def_name}/default/{job_id}"
 
@@ -308,6 +309,7 @@ def test_cancel_pending_job():
     resp = batch_client.describe_jobs(jobs=[job_id])
     resp["jobs"][0]["jobName"].should.equal("test_job_name")
     resp["jobs"][0]["statusReason"].should.equal("test_cancel")
+    resp["jobs"][0]["container"].shouldnt.have.key("logStreamName")
 
 
 @mock_logs
@@ -341,6 +343,7 @@ def test_cancel_running_job():
     resp = batch_client.describe_jobs(jobs=[job_id])
     resp["jobs"][0]["jobName"].should.equal("test_job_name")
     resp["jobs"][0].shouldnt.have.key("statusReason")
+    resp["jobs"][0]["container"].should.have.key("logStreamName")
 
 
 @mock_batch
@@ -399,7 +402,7 @@ def test_failed_job():
     _, _, _, iam_arn = _setup(ec2_client, iam_client)
 
     job_def_name = "exit-1"
-    commands = ["exit", "1"]
+    commands = ["kill"]
     job_def_arn, queue_arn = prepare_job(batch_client, commands, iam_arn, job_def_name)
 
     resp = batch_client.submit_job(
@@ -413,6 +416,7 @@ def test_failed_job():
         resp = batch_client.describe_jobs(jobs=[job_id])
 
         if resp["jobs"][0]["status"] == "FAILED":
+            resp["jobs"][0]["container"].should.have.key("logStreamName")
             break
         if resp["jobs"][0]["status"] == "SUCCEEDED":
             raise RuntimeError("Batch job succeeded even though it had exit code 1")
@@ -552,7 +556,7 @@ def test_failed_dependencies():
             "image": "busybox:latest",
             "vcpus": 1,
             "memory": 128,
-            "command": ["exit", "1"],
+            "command": ["kill"],
         },
     )
     job_def_arn_failure = resp["jobDefinitionArn"]
@@ -592,6 +596,13 @@ def test_failed_dependencies():
         assert resp["jobs"][1]["status"] != "SUCCEEDED", "Job 3 cannot succeed"
 
         if resp["jobs"][1]["status"] == "FAILED":
+            assert resp["jobs"][0]["container"].should.have.key(
+                "logStreamName"
+            ), "Job 2 should have logStreamName because it FAILED but was in RUNNING state"
+            assert resp["jobs"][1]["container"].shouldnt.have.key(
+                "logStreamName"
+            ), "Job 3 shouldn't have logStreamName because it was never in RUNNING state"
+
             break
 
         time.sleep(0.5)
diff --git a/tests/test_batch_simple/test_batch_jobs.py b/tests/test_batch_simple/test_batch_jobs.py
index d797a3320..03c945dde 100644
--- a/tests/test_batch_simple/test_batch_jobs.py
+++ b/tests/test_batch_simple/test_batch_jobs.py
@@ -70,6 +70,7 @@ def test_submit_job_by_name():
     job["status"].should.equal("SUCCEEDED")
     job.should.contain("container")
     job["container"].should.contain("command")
+    job["container"].should.contain("logStreamName")
 
 
 @mock_batch_simple