Database outsourcing is emerging as an important trend in which the data owners ship their data to external service provider. Since a service provider is not fully trusted, security and privacy are significant issues. They can be referred to as data confidentiality, user and data privacy, query assurance and secure auditing. Among them, secure auditing takes an important role to have a secure database outsourcing model. While audit log provides investigator information to assess past system activity, it often contains sensitive information. We need to protect it from unauthorized parties while making it still efficiently searchable by authorized ones. In this paper, we propose a novel searchable encryption scheme, named PEKS-PM, based on pseudo inverse matrix, which enables a trusted party to give the service provider's server the ability to test whether a given keyword appears in log records but the server learns nothing about the keyword and the log content. Our encryption scheme is based mainly on linear matrix operations therefore the complexity is low. We prove that our proposed encryption scheme is secure and efficient, comparing with an encryption scheme based on the Decisional Diffie-Hellman (DDH) assumption.