Example 1

On applying prop. 6 of V, p. 230, to the relation $1/x \ll x^{\alpha-1}$ where $\alpha > 0$, one again obtains the relation $\log x \ll x^{\alpha}$ for every $\alpha > 0$, which is equivalent to the relation $y^{1/\alpha} \ll e^y$ proved in III, p. 105.