Let's take $a^{n}$ = $2^{8}$ for simplicity.
now max. no of multiplication required is = 2X2X2X2X2X2X2X2 = 256 5 multiplications or we can say (n-1) multiplications.
min. no of multiplications required = 2X2=4
=$2^{2}$X$2^{2}$=16
=$2^{4}$X$2^{4}$=256 Log n multiplications.
We are multiplying the result of one multiplication to itself and again doing the same, that reduces the no. of multiplications required.