We give some sufficient conditions on distribution \( |\{x\in \Omega :\,p(x)\leq 1+\lambda\}|\,(\lambda>0) \) of the exponent function \( p(\cdot): \Omega \to [0,\infty) \) that implies the embedding \( L^{p(\cdot)}(\Omega)\subset L(LogL)^{\alpha}(\Omega)\,(\alpha>0) \), where \( \Omega \) is an open set with finite Lebesgue measure.  We show that this result is in a sense sharp. As an application of our approach we give conditions that imply the strong differentiation of integrals of functions from \( L^{p(\cdot)}((0,1)^{n})\,(n>1) \) and integrability of maximal function for variable Lebesgue spaces where the exponent function \( p(\cdot) \) approaches 1 in value on some part of domain.