There is a proof (Theorem 1.2) in I. Csiszár and P.C. Shields (2004) with a line I don't understand. This is the setup:
Thus $m \geq 1, 0 \leq r < 2^{m+1}$.
We have to prove that $m + r2^{-(m+1)} \leq \log_2(2^{m+1} - 2 + r)$. (Statement 1)
or $\quad r2^{-(m+1)} \leq \log_2(2 + (r-2)2^{-m})$ (Statement 2)
How is proving statement 1 equivalent to proving statement 2? Thank you.
I. Csiszár and P.C. Shields (2004), "Information Theory and Statistics: A Tutorial", Foundations and Trends® in Communications and Information Theory: Vol. 1: No. 4, pp 417-528. http://dx.doi.org/10.1561/0100000004
I found the trick they use.
$\log_2(2^{-n} x) = -n + \log_2(x)$.
So,
\begin{align} \log_2(2^{m+1} + r) - m &= \log_2(2^{-m} (2^{m+1} - 2 + r)) \\ &= \log_2(2 - 2^{-m+1} + r2^{-m}) \\ &= \log_2(2 + (r-2) 2^{-m}) \end{align}