In this paper, we propose a bit-depth scalable lossless coding method for high dynamic range (HDR) images based on a reversible logarithmic mapping. HDR images are generally expressed as floating-point data, such as in the OpenEXR or RGBE formats. Our bit-depth scalable coding approach outputs base layer data and enhancement layer data. It can reconstruct the low dynamic range (LDR) image from the base layer data and reconstructs the HDR image by adding the enhancement layer data. Most previous two-layer methods have focused on the lossy coding of HDR images. Unfortunately, the extension of previous lossy methods to lossless coding does not significantly compress the enhancement layer data. This is because the bit depth becomes very large, especially for HDR images in floating-point data format. To tackle this problem, we apply a reversible logarithmic mapping to the input HDR data. Moreover, we introduce a format conversion to avoid any degradation in the quality of the reconstructed LDR image. The proposed method is effective for both OpenEXR and RGBE formats. Through a series of experiments, we confirm that the proposed method decreases the volume of compressed data while maintaining the visual quality of the reconstructed LDR images.