My doubt is in a step of the following lemma
Lemma Let w be a non-decreasing function on an interval (0, Ro] satisfying, for all $R \le R_0$, the inequality \begin{equation} w(\tau R) \le \gamma w(R) + \sigma(R) \end{equation} where $\sigma$ is also non-decreasing and $0 < \gamma, \tau< 1$. Then, for any $ \mu \in (0, 1)$ and $R \le R_0$, we have \begin{equation} w(R) \le C \left(\left(\frac{R}{R_0} \right)^\alpha w(R_0) + \sigma (R^\mu R^{1-\mu}) \right) .\end{equation} where $c= C( \gamma, \tau)$ and $\alpha = \alpha(\gamma, \tau ,\mu)$ are positive constants.
The doubt that I have is in the following step of the short proof of the lemma. For any $ R \le R_1, m$ was chosen so that $$ \tau^m R_1 \le R \le \tau^{m-1} R_1 $$ How can I see that $$ \gamma^{m-1} \le \frac{1}{\gamma} \left (\frac{R}{R_1} \right )^{\log \gamma /\log \tau}? $$ My thoughts $$ \gamma^{m-1} = \frac{1}{\gamma} \gamma^m $$ and $$ \tau^m < R/R_1$$ It follows that (as $ \gamma <1$) $$ m> \log_\tau \left (R/R_1\right ) $$
This is indeed very simple: You want
$$\gamma^{m} \le \left (\frac{R}{R_1} \right )^{\log \gamma /\log \tau}, $$
taking $\log$, this is equivalent to
\begin{align} m \log \gamma &\le \frac{\log \gamma}{\log \tau} \log\left (\frac{R}{R_1} \right )\\ \Leftrightarrow m \log \tau &\le \log\frac{R}{R_1} \qquad \qquad \ \ \left( \text{note }\frac{\log \gamma}{\log \tau}>0\right)\\ \Leftrightarrow \tau ^m &\le \frac{R}{R_1} \end{align}
which is $\tau^m R_1\le R$.