This mini-survey summarises key ideas used in proofs of the best known (till 2007) estimates for the irrationality exponents of $log2$ [*E. A. Rukhadze*, Mosc. Univ. Math. Bull. 42, No. 6, 30–35 (1987); translation from Vestn. Mosk. Univ., Ser. I 1987, No. 6, 25–29 (1987; Zbl 0635.10025)], $\pi $ [*M. Hata*, Acta Arith. 63, No. 4, 335–349 (1993; Zbl 0776.11033)], and $log3$ [*G. Rhin*, Théorie des nombres, Sémin. Paris 1985/86, Prog. Math. 71, 155–164 (1987; Zbl 0632.10034)].

Recall that the irrationality exponent $\mu =\mu \left(\gamma \right)$ of a real irrational number $\gamma $ is the infimum of quantities $c\in \mathbb{R}$, for which the inequality ${|\gamma -p/q|<\left|q\right|}^{-c}$ has only finitely many solutions in integers $p$ and $q\ne 0$.

It seems quite remarkable that the long-standing records of Hata and Rhin have been recently broken by Salikhov, who proves the estimates $\mu \left(\pi \right)<7\xb760630853$ [*V. Kh. Salikhov*, Russ. Math. Surv. 63, No. 3, 570–572 (2008); translation from Usp. Mat. Nauk 63, No. 3, 163–164 (2008; Zbl 1208.11086)] and $\mu (log3)<5\xb7125$ [*V. Kh. Salikhov*, Dokl. Math. 76, No. 3, 955-957 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 417, No. 6, 753–755 (2007; Zbl 1169.11032)], while *R. Marcovecchio* announces a considerable improvement of Rukhadze’s estimate for $log2$; his new mark is $\mu (log2)<3\xb757455391$ [Acta Arith. 139, No. 2, 147–184 (2009; Zbl 1197.11083)].

Reviewer’s remark: The author’s translation of the article under review into English can be found at

##### MSC:

11J82 | Measures of irrationality and of transcendence |

11J72 | Irrationality |

11J91 | Transcendence theory of other special functions |