This study presents a novel numerical approach that improves the training efficiency of binary logistic regression, a popular statistical model in the machine learning community. Our method achieves training times an order of magnitude faster than traditional logistic regression by employing a novel Soft-Plus approximation, which enables reformulation of logistic regression parameter estimation into matrix-vector form. We also adopt the Lf-norm penalty, which allows using fractional norms, including the L2-norm, L1-norm, and L0-norm, to regularize the model parameters. We put Lf-norm formulation in matrix-vector form, providing flexibility to include or exclude penalization of the intercept term when applying regularization. Furthermore, to address the common problem of collinear features, we apply singular value decomposition (SVD), resulting in a low-rank representation commonly used to reduce computational complexity while preserving essential features and mitigating noise. Moreover, our approach incorporates a randomized SVD alongside a newly developed SVD with row reduction (SVD-RR) method, which aims to manage datasets with many rows and features efficiently. This computational efficiency is crucial in developing a generalized model that requires repeated training over various parameters to balance bias and variance. We also demonstrate the effectiveness of our fast binary logistic regression (FBLR) method on various datasets from the OpenML repository in addition to synthetic datasets.