Let f be an E-function (in Siegel’s sense) not of the form \(e^{\beta z}\) , \(\beta \in \overline{\mathbb {Q}}\) , and let \(\log \) denote any fixed determination of the complex logarithm. We first prove that there exists a finite set S(f) such that for all \(\xi \in \overline{\mathbb {Q}}\setminus S(f)\) , \(\log (f(\xi ))\) is a transcendental number. We then quantify this result when f is an E-function in the strict sense with rational coefficients, by proving an irrationality measure of \(\ln (f(\xi ))\) when \(\xi \in \mathbb Q\setminus S(f)\) and \(f(\xi )>0\) . This measure implies that \(\ln (f(\xi ))\) is not an ultra-Liouville number, as defined by Marques and Moreira. The proof of our first result, which is in fact more general, uses in particular a recent theorem of Delaygue. The proof of the second result, which is independent of the first one, is a consequence of a new linear independence measure for values of linearly independent E-functions in the strict sense with rational coefficients, where emphasis is put on other parameters than on the height, contrary to the case in Shidlovskii’s classical measure for instance.
