Log function properties and time series data

(The formulas you wrote are not exactly true, but approximately true.)

Your second formula comes from the fact that $s\ln{x} = \ln\left(x^{s}\right) \sim x^{s}-1$ as $x\to 1$, for any $s\in \Bbb{R}$.

Therefore, if $\frac{y_{t+1}}{y_{t}}\approx 1$, we will have $\left(\frac{y_{t+1}}{y_{t}}\right)^{s}-1\approx s\ln \left(\frac{y_{t+1}}{y_{t}}\right) = s\left(\ln \left(y_{t+1}\right) - \ln \left(y_{t}\right)\right) $.

To see that $\ln\left(x^{s}\right) \sim x^{s}-1$ as $x\to 1$ from the fact that $\ln u \sim u-1$ as $u\to 1$, basically just make the substitution $u=x^{s}$. That is, note that for any $s\in \Bbb{R}$, we have $$\begin{align} \lim\limits_{x\to 1} \frac{\ln\left(x^{s}\right)}{x^{s}-1} &= \lim\limits_{u\to 1}\frac{\ln u}{u-1} = 1. \end{align}$$