Question

Find the largest integer n for which n power 200 < 5 power 300?

Answer 1

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

Answer 2

`n^(200)<5^(300)\\\\(n^2)^(100)<(5^3)^(100)\ \ \ \Rightarrow\ \ \ n^2<5^3\ \ \ \Rightarrow\ \ \ n^2<125\\\\n\in I\ \ \ \Rightarrow\ \ \ n< √(125) \ \ \ and\ \ \ √(125) =5 √(5) \approx11.18\\\\Ans.\ the\ largest\ integer\ n\ is\ 11`