In the book " Number Theory IV Transcendental Numbers" written by Parsin and Shafarevich ([book](https://link.springer.com/book/10.1007/978-3-662-03644-0), page 104) it is asserted that to explicit a transcendence measure of a complex number $w$, it is sufficient to prove that there is an infinite sequence of polynomials $P\_m(x)\in\mathbb Z[x]$ of fixed degree such that
$$0<H(P\_m)\le c^m\qquad e^{-\lambda\_1m}\le|P\_m(w)[\le e^{-\lambda\_2m}$$
where $c$, $\lambda\_1$, and $\lambda\_2$ are constants, such that $c > 1, \lambda\_l > \lambda\_2 > 0$ with $H(P\_m)=\max\{|\text{coefficients of $P\_m$}|\}$. I do not know why. Can anyone give the argument behind this assertion or a reference?
This result is used to give a transcendence measure of $\ln(r)$ ($r$ is a rational)


Thanks in advance



Transcendence measure: of $\ln(a/b)$
Warning: this is for irrationaity measure, not transcendence measure.


Let $a/b$ be an approximation of $w$ such that $|w-a/b|=b^{-\kappa}$. Then $$P\_m(a/b)=P\_m(w)+(w-a/b)P\_m'(\theta)$$
for certain $\theta$ between $a/b$ and $w$. Note that $P\_m(a/b)$ is either 0 or at least $b^{-d}$ in absolute value, where $\deg P\_m\leqslant d$. Choose $m$ such that $b^{-d}\geqslant 2e^{-\lambda\_2 m}$, say, $m=\lceil \frac{\log 2+d\log b}{\lambda\_2}\rceil$.


Then, if $|P\_m(a/b)|\geqslant b^{-d}$, we get $$C(d,w)\cdot c^m b^{-\kappa}\geqslant |b^{-\kappa}P\_m'(\theta)|=|(w-a/b)P\_m'(\theta)|=|P\_m(a/b)-P\_m(w)|\geqslant e^{-\lambda\_2 m},$$
thus $\kappa\log b\leqslant m(\log c+\lambda\_2)+O(1)$ and $\kappa\leqslant d(1+\frac{\log c}{\lambda\_2})+o(1)$.


If $P\_m(a/b)=0$, then analogously
$$C(d,w)\cdot c^m b^{-\kappa}\geqslant |b^{-\kappa}P\_m'(\theta)|=|(w-a/b)P\_m'(\theta)|=|P\_m(w)|\geqslant e^{-\lambda\_1 m},$$
so $\kappa\log b\leqslant m(\log c+\lambda\_1)+O(1)$, and $\kappa\leqslant d\cdot \frac{\log c+\lambda\_1}{\lambda\_2}+o(1)$.


So, in both cases we may conclude that the irrationality measure of $w$ does not exceed $$d\cdot \frac{\log c+\lambda\_1}{\lambda\_2}.$$


