Let us take logarithms on both sides. 200 log n < 300 log 5 So log n < 3/2 log 5 log n < log 5 power 3/2 n < 5 power 3/2 n < square root (5³) = √125
so n = 11
n 200 < 5 300 ( n 2 ) 100 < ( 5 3 ) 100 ⇒ n 2 < 5 3 ⇒ n 2 < 125 ∈ I ⇒ n < 125 an d 125 = 5 5 ≈ 11.18 A n s . t h e l a r g es t in t e g er n i s 11
The largest integer n such that n 200 < 5 300 is 11, derived by manipulating and simplifying the inequality. This is determined by taking the 200th root of both sides and evaluating that result. Hence, n < 11.18 , and the maximum integer value is 11.
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