  The $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the
$q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$
($x\otimes_1 y=x y$) emerge naturally within nonextensive statistical
mechanics. We show here how they lead to two-parameter (namely, $q$ and
$q^\prime$) generalizations of the logarithmic and exponential functions (noted
respectively $\ln_{q,q^\prime}x$ and $e_{q,q^\prime}^{x}$), as well as of the
Boltzmann-Gibbs-Shannon entropy $S_{BGS}\equiv -k \sum_{i=1}^Wp_i \ln p_i$
(noted $S_{q,q^\prime}$). The remarkable properties of the
$(q,q^\prime)$-generalized logarithmic function make the entropic form
$S_{q,q^\prime} \equiv k \sum_{i=1}^W p_i \ln_{q,q^\prime}(1/p_i)$ to satisfy,
for large regions of $(q,q^\prime)$, important properties such as {\it
expansibility}, {\it concavity} and {\it Lesche-stability}, but not necessarily
{\it composability}.
