Add logging for user-assigned managed identity (#44305)

Fixes #43741

Author: billwert

sdk/identity/azure-identity/src/main/java/com/azure/identity/ManagedIdentityCredential.java

@@ -199,6 +199,7 @@ public Mono<AccessToken> getToken(TokenRequestContext request) {
                     + " https://aka.ms/azsdk/java/identity/managedidentitycredential/troubleshoot")));
         }
 
+        // Not having a managedIdentityId at this point means it is a system-assigned managed identity.
         if (!CoreUtils.isNullOrEmpty(managedIdentityId)) {
             ManagedIdentitySourceType managedIdentitySourceType = ManagedIdentityApplication.getManagedIdentitySource();
             if (ManagedIdentitySourceType.CLOUD_SHELL.equals(managedIdentitySourceType)
@@ -211,6 +212,8 @@ public Mono<AccessToken> getToken(TokenRequestContext request) {
                             ? "DefaultAzureCredentialBuilder."
                             : "ManagedIdentityCredentialBuilder."))));
             }
+
+            LOGGER.info("User-assigned Managed Identity ID: " + getClientId());
         }
 
         return managedIdentityServiceCredential.authenticate(request)