Tight lower and upper bounds on the ratio of relative entropies of two probability distributions with respect to a common third one are established, where the three distributions are collinear in the standard $(n-1)$-simplex. These bounds are leveraged to analyze the capacity of an arbitrary binary-input channel with noisy causal state information (provided by a side channel) at the encoder and perfect state information at the decoder, and in particular to determine the exact universal threshold on the noise measure of the side channel, above which the capacity is the same as that with no encoder side information.