  We shall present effective approximations measures for certain infinite
products related to $q$-exponential function. There are two main targets. First
we shall prove an explicit irrationality measure result for the values of
$q$-exponential function at rational points. Then, if we restrict the
approximations to rational numbers of the shape $d^s/N$, we may replace
Bundschuh's irrationality exponent $7/3$ by $2+\frac{1}{3+2\sqrt 3}=2.1547...$.
