  We have obtained the optimal upper bound of entropic uncertainty relation for
$N$ Mutually Unbiased Bases (MUBs). We have used the methods of variational
calculus for the states that can be written in terms of $N$ MUBs. Our result is
valid for any state when $N$ is $d+1$, where $d$ is the dimension of the
related system. We provide a quantitative criterion for the extendibilty of
MUBs. In addition, we have applied our result to the mutual information of
$d+1$ observables conditioned with a classical memory.
