# Reprint: On the approximation of the exponential function and the logarithm. I, II (1931/32)

### Kurt Mahler

## Abstract

In Part I, Mahler introduces his classification of complex numbers and the following two results are proved. Let $ϑ_{1},ϑ_{2},…,ϑ_{N}$ be $N$ algebraic numbers that are linearly independent over the rationals and let $λ$ be a Liouville number. Then, the numbers $e_{ϑ_{1}},e_{ϑ_{2}},…,e_{ϑ_{N}},λ$ are algebraically independent over the field of algebraic numbers. Let $z$ be the real logarithm of a positive rational number not equal to one and let $λ$ be a Liouville number. Then, $z$ and $λ$ are algebraically independent over the field of algebraic numbers.

Part II continues the study of the same title by giving various bounds on polynomials evaluated at logarithms and exponentials. A new proof of the transcendence of $π$ is given as an application.

Reprint of the author's papers [J. Reine Angew. Math. 166, 118--136 (1931; Zbl 0003.15101; JFM 57.0242.03); ibid. 166, 137--150 (1932; Zbl 0003.38805; JFM 58.0207.01)].