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

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Kate200468id
[tex]n^{200}<5^{300}\\\\(n^2)^{100}<(5^3)^{100}\ \ \ \Rightarrow\ \ \ n^2<5^3\ \ \ \Rightarrow\ \ \ n^2<125\\\\n\in I\ \ \ \Rightarrow\ \ \ n< \sqrt{125} \ \ \ and\ \ \ \sqrt{125} =5 \sqrt{5} \approx11.18\\\\Ans.\ the\ largest\ integer\ n\ is\ 11[/tex]
Everest2017id
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