Sonic and bulk density logs are crucial inputs for many subsurface tasks including formation identification, completion design, and porosity estimation. Economic and operational concerns restrict the acquisition of these logs, meaning the overburden and sometimes entire wells are completely unlogged. In contrast, parameters that monitor drilling operations, such as weight on bit and torque, are recorded for every borehole. Previous studies have applied supervised machine learning approaches to predict these missing logs from the drilling parameters. While the results are promising, they often do not investigate the importance of different features and the corresponding practical implications. Here, we explored the feasibility of predicting compressional slowness and bulk density logs using various combinations of formation markers, gamma-ray logs, and drilling data recorded at the rig. Our tests utilized a temporal convolutional network to allow the model to learn from sequences of input features. Bayesian-based hyperparameter tuning found the optimum set of parameters for each experiment before producing the final log predictions. Finally, a permutation feature importance analysis revealed which input variables contributed most to the outputs. Although drilling parameters contain some insight into the mechanical rock properties, we found that they cannot produce the high-quality log predictions required for many tasks. Supplementing the drilling parameters with a gamma-ray log and formation data produces good-quality log predictions, with the additional inputs helping to constrain the model outputs.