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Logical queries constitute an important subset of questions posed in knowledge graph question answering systems. Yet, effectively answering logical queries on large knowledge graphs remains a highly challenging problem. Traditional subgraph matching based methods might suffer from the noise and incompleteness of the underlying knowledge graph, often with a prolonged online response time. Recently, an alternative type of method has emerged whose key idea is to embed knowledge graph entities and the query in an embedding space so that the embedding of answer entities is close to that of the query. Compared with subgraph matching based methods, it can better handle the noisy or missing information in knowledge graph, with a faster online response. Promising as it might be, several fundamental limitations still exist, including the linear transformation assumption for modeling relations and the inability to answer complex queries with multiple variable nodes. In this paper, we propose an embedding based method (NewLook) to address these limitations. Our proposed method offers three major advantages. First (Applicability), it supports four types of logical operations and can answer queries with multiple variable nodes. Second (Effectiveness), the proposed NewLook goes beyond the linear transformation assumption, and thus consistently outperforms the existing methods. Third (Efficiency), compared with subgraph matching based methods, NewLook is at least 3 times faster in answering the queries; compared with the existing embed-ding based methods, NewLook bears a comparable or even faster online response and offline training time.