What is the smallest integer n for which…
To solve, we want to get the bases the same. Thus we need to break 25^n into prime factors.
25^n = (5^2)^n = 5^(2n) (Remember that when we have a power to a power, we multiply the exponents.)
We can use the new value in the given inequality:
Since we have the same bases on either side of the inequality we can drop the bases and set up an equation involving just the exponents.
2n > 12
n > 6
Because n is greater than 6, the smallest integer that satisfies the inequality 25^n > 5^12 is 7.