diff --git a/sentry_sdk/_log_batcher.py b/sentry_sdk/_log_batcher.py
index 87bebdb226..f7f6c80565 100644
--- a/sentry_sdk/_log_batcher.py
+++ b/sentry_sdk/_log_batcher.py
@@ -13,15 +13,18 @@
 
 class LogBatcher:
     MAX_LOGS_BEFORE_FLUSH = 100
+    MAX_LOGS_BEFORE_DROP = 1_000
     FLUSH_WAIT_TIME = 5.0
 
     def __init__(
         self,
         capture_func,  # type: Callable[[Envelope], None]
+        record_lost_func,  # type: Callable[..., None]
     ):
         # type: (...) -> None
         self._log_buffer = []  # type: List[Log]
         self._capture_func = capture_func
+        self._record_lost_func = record_lost_func
         self._running = True
         self._lock = threading.Lock()
 
@@ -79,6 +82,14 @@ def add(
             return None
 
         with self._lock:
+            if len(self._log_buffer) >= self.MAX_LOGS_BEFORE_DROP:
+                self._record_lost_func(
+                    reason="queue_overflow",
+                    data_category="log_item",
+                    quantity=1,
+                )
+                return None
+
             self._log_buffer.append(log)
             if len(self._log_buffer) >= self.MAX_LOGS_BEFORE_FLUSH:
                 self._flush_event.set()
diff --git a/sentry_sdk/client.py b/sentry_sdk/client.py
index 6cb5ca5826..2c245297bd 100644
--- a/sentry_sdk/client.py
+++ b/sentry_sdk/client.py
@@ -386,7 +386,10 @@ def _record_lost_event(
             if has_logs_enabled(self.options):
                 from sentry_sdk._log_batcher import LogBatcher
 
-                self.log_batcher = LogBatcher(capture_func=_capture_envelope)
+                self.log_batcher = LogBatcher(
+                    capture_func=_capture_envelope,
+                    record_lost_func=_record_lost_event,
+                )
 
             self.metrics_batcher = None
             if has_metrics_enabled(self.options):
diff --git a/tests/test_logs.py b/tests/test_logs.py
index da7d0d5f03..6c0a9b14f9 100644
--- a/tests/test_logs.py
+++ b/tests/test_logs.py
@@ -446,3 +446,28 @@ def test_logs_with_literal_braces(
         assert logs[0]["attributes"]["sentry.message.template"] == message
     else:
         assert "sentry.message.template" not in logs[0]["attributes"]
+
+
+@minimum_python_37
+def test_batcher_drops_logs(sentry_init, monkeypatch):
+    sentry_init(enable_logs=True)
+    client = sentry_sdk.get_client()
+
+    def no_op_flush():
+        pass
+
+    monkeypatch.setattr(client.log_batcher, "_flush", no_op_flush)
+
+    lost_event_calls = []
+
+    def record_lost_event(reason, data_category=None, item=None, *, quantity=1):
+        lost_event_calls.append((reason, data_category, item, quantity))
+
+    monkeypatch.setattr(client.log_batcher, "_record_lost_func", record_lost_event)
+
+    for i in range(1_005):  # 5 logs over the hard limit
+        sentry_sdk.logger.info("This is a 'info' log...")
+
+    assert len(lost_event_calls) == 5
+    for lost_event_call in lost_event_calls:
+        assert lost_event_call == ("queue_overflow", "log_item", None, 1)