Transcendence of values of logarithms of E-functions

Let f be an E-function (in Siegel's sense) not of the form e^ z, Q, and let denote any fixed determination of the complex logarithm. We first prove that there exists a finite set S (f) such that for all Q S (f), (f () ) 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 (f () ) when Q S (f) and f () 0. This measure implies that (f () ) 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.