Lower Bound on Log function

Question

In this paper "Papandriopoulos, J.; Evans, J.S., "SCALE: A Low-Complexity Distributed Protocol for Spectrum Balancing in Multiuser DSL Networks," Information Theory, IEEE Transactions on , vol.55, no.8, pp.3711,3724, Aug. 2009"

The authors used the following lower bound on the log function: \begin{equation} \alpha \log z + \beta \leq \log (1+z) \end{equation} that is tight when $z=z_o$ when the approximation constants are chosen as

\begin{equation} \alpha=\frac{z_o}{1+z_o}, ~ \beta=\log(1+z_o)-\frac{z_o}{1+z_o}\log(z_o) \end{equation}

However, it is not mentioned in this paper how those constants were derived as mentioned. I don't think they are chosen arbitrarily. So, any ideas how $\alpha$ and $\beta$ were chosen like that ?

Answer 1

The constants are chosen such that the bound $\alpha \log z + \beta$ has the same value and first derivative as $\log (1+z)$ in $z_0$. That makes it the best possible bound of that form near $z_0$.