  We show how the theory of linear forms in two logarithms allows one to get
effective irrationality measures for $n$-th roots of rational numbers ${a \over
b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result
under the assumption that $a$ is $p$-adically very close to $b$, that is, that
a large power of $p$ divides $a-b$. As an application, we solve completely
certain families of Thue-Mahler equations. Our results illustrate, admittedly
in a very special situation, the strength of the known estimates for linear
forms in logarithms.
