  We give a sharp sufficient condition on the distribution function, $|\{x\in
\Omega :\,p(x)\leq 1+\lambda\}|$, $\lambda>0$, of the exponent function
$p(\cdot): \Omega \to [1,\infty)$ that implies the embedding of the variable
Lebesgue space $L^{p(\cdot)}(\Omega)$ into the Orlicz space $L(\log
L)^{\alpha}(\Omega)$, $\alpha>0$, where $\Omega$ is an open set with finite
Lebesgue measure. As applications of our results, we first give conditions that
imply the strong differentiation of integrals of functions in
$L^{p(\cdot)}((0,1)^{n})$, $n>1$. We then consider the integrability of the
maximal function on variable Lebesgue spaces, where the exponent function
$p(\cdot)$ approaches $1$ in value on some part of the domain. This result is
an improvement of the result in~\cite{CUF2}.
