Various factorization-based methods have been proposed to leverage second-order, or higher-order cross features for boosting the performance of predictive models. They generally enumerate all the cross features under a predefined maximum order, and then identify useful feature interactions through model training, which suffer from two drawbacks. First, they have to make a trade-off between the expressiveness of higher-order cross features and the computational cost, resulting in suboptimal predictions. Second, enumerating all the cross features, including irrelevant ones, may introduce noisy feature combinations that degrade model performance. In this work, we propose the Adaptive Factorization Network (AFN), a new model that learns arbitrary-order cross features adaptively from data. The core of AFN is a logarithmic transformation layer that converts the power of each feature in a feature combination into the coefficient to be learned. The experimental results on four real datasets demonstrate the superior predictive performance of AFN against the state-of-the-arts.