Transcendence measure: of $\ln(a/b)$ In the book " Number Theory IV Transcendental Numbers" written by Parsin and Shafarevich (book, 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
 A: 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 $$d\cdot \frac{\log c+\lambda_1}{\lambda_2}.$$
