In this paper, we present an energy-efficient SNN architecture, which can seamlessly run deep spiking neural networks (SNNs) with improved accuracy. First, we propose a conversion aware training (CAT) to reduce ANN-to-SNN conversion loss without hardware implementation overhead. In the proposed CAT, the activation function developed for simulating SNN during ANN training, is efficiently exploited to reduce the data representation error after conversion. Based on the CAT technique, we also present a time-to-first-spike coding that allows lightweight logarithmic computation by utilizing spike time information. The SNN processor design that supports the proposed techniques has been implemented using 28nm CMOS process. The processor achieves the top-1 accuracies of 91.7%, 67.9% and 57.4% with inference energy of 486.7uJ, 503.6uJ, and 1426uJ to process CIFAR-10, CIFAR-100, and Tiny-ImageNet, respectively, when running VGG-16 with 5bit logarithmic weights.