The tensor exponential function has been widely used in cybernetics, image processing and various engineering fields. Based on the generalized matrix inverse, an effective tensor generalized inverse is defined for the first time on the scalar inner product space, thus constructing a continued fractional algorithm for the tensor Pad$\acutee$ approximation. On the other hand, we successfully use the tensor t- product to calculate the power of the tensor, and recursively giving the power series expansion of the tensor exponential function. Based on the previous two work, the continuous fractional algorithm designed in this paper is used to approximate the tensor exponential function. Its characteristic is that the algorithm can be programmed to implement recursive calculations, and in the calculation process, it is not necessary to calculate the product of the tensor and to calculate the inverse of the tensor. The numerical experiments of the two tensor exponential functions given in this paper show that comparing the continuous fractional algorithm with the commonly used truncation method, the proposed algorithm is effective without reducing the approximation order. If the dimension of the tensor is relatively large, a continuous fractional algorithm based on the generalized inverse of tensors will also have certain advantages.