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.