CREATE TABLE DBO.SynonymTriggerLogging( ID int Identity(1,1) PRIMARY KEY NOT NULL,
                                 LoginName varchar(200),
                                 QueryExecuted varchar(max));

CREATE TRIGGER SynonymTrigger
ON DATABASE
AFTER DROP_SYNONYM
AS
INSERT INTO DBO.SynonymTriggerLogging(LoginName,QueryExecuted)
SELECT ORIGINAL_LOGIN(),EVENTDATA().value('(/EVENT_INSTANCE/TSQLCommand/CommandText)[1]','nvarchar(max)');

CREATE SYNONYM TEST
FOR dbo.test;

DROP SYNONYM TEST;

SELECT *
FROM DBO.SynonymTriggerLogging;

ID  LoginName   QueryExecuted
1   rvertongen  DROP SYNONYM TEST

CREATE TABLE DBO.SynonymTriggerLogging( ID int Identity(1,1) PRIMARY KEY NOT NULL,
                                 LoginName varchar(200),
                                 QueryExecuted varchar(max),
                                 HostName varchar(200));
GO
CREATE TRIGGER SynonymTrigger
ON DATABASE
FOR DROP_SYNONYM
AS
INSERT INTO DBO.SynonymTriggerLogging(LoginName,QueryExecuted,HostName)
SELECT ORIGINAL_LOGIN(),EVENTDATA().value('(/EVENT_INSTANCE/TSQLCommand/CommandText)[1]','nvarchar(max)'),HOST_NAME()