diff --git a/sentry_sdk/client.py b/sentry_sdk/client.py
index 4dfccb3132..102392c61d 100644
--- a/sentry_sdk/client.py
+++ b/sentry_sdk/client.py
@@ -27,6 +27,7 @@
 from sentry_sdk.tracing import trace
 from sentry_sdk.transport import BaseHttpTransport, make_transport
 from sentry_sdk.consts import (
+    SPANDATA,
     DEFAULT_MAX_VALUE_LENGTH,
     DEFAULT_OPTIONS,
     INSTRUMENTER,
@@ -894,6 +895,10 @@ def _capture_experimental_log(self, current_scope, log):
             return
         isolation_scope = current_scope.get_isolation_scope()
 
+        server_name = self.options.get("server_name")
+        if server_name is not None and SPANDATA.SERVER_ADDRESS not in log["attributes"]:
+            log["attributes"][SPANDATA.SERVER_ADDRESS] = server_name
+
         environment = self.options.get("environment")
         if environment is not None and "sentry.environment" not in log["attributes"]:
             log["attributes"]["sentry.environment"] = environment
diff --git a/tests/test_logs.py b/tests/test_logs.py
index fb824760a8..d58aa9acdd 100644
--- a/tests/test_logs.py
+++ b/tests/test_logs.py
@@ -11,6 +11,7 @@
 from sentry_sdk.envelope import Envelope
 from sentry_sdk.integrations.logging import LoggingIntegration
 from sentry_sdk.types import Log
+from sentry_sdk.consts import SPANDATA
 
 minimum_python_37 = pytest.mark.skipif(
     sys.version_info < (3, 7), reason="Asyncio tests need Python >= 3.7"
@@ -161,7 +162,7 @@ def test_logs_attributes(sentry_init, capture_envelopes):
     """
     Passing arbitrary attributes to log messages.
     """
-    sentry_init(_experiments={"enable_logs": True})
+    sentry_init(_experiments={"enable_logs": True}, server_name="test-server")
     envelopes = capture_envelopes()
 
     attrs = {
@@ -184,6 +185,7 @@ def test_logs_attributes(sentry_init, capture_envelopes):
     assert logs[0]["attributes"]["sentry.environment"] == "production"
     assert "sentry.release" in logs[0]["attributes"]
     assert logs[0]["attributes"]["sentry.message.parameters.my_var"] == "some value"
+    assert logs[0]["attributes"][SPANDATA.SERVER_ADDRESS] == "test-server"
 
 
 @minimum_python_37