If $2^{n-7} \times 5^{n-4} = 1250$, find n.

Finding prime factors of 1250,

$2^{n-7} \times 5^{n-4} = 1250\\[1em] \Rightarrow 2^{n-7} \times 5^{n-4} = 2^1 \times 5^4$

On comparing the exponent of 2 or 5, we get

$n - 7 = 1 \\[1em] \Rightarrow n = 1 + 7\\[1em] \Rightarrow n = 8$

OR

$n - 4 = 4\\[1em] \Rightarrow n = 4 + 4\\[1em] \Rightarrow n = 8$

If $2^{n-7} \times 5^{n-4} = 1250$, then $n = 8$.