# arrow_back What is the maximum possible value of $n$ in the inequality $n^{200}<5^{300}$?

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What is the maximum possible value of $n$ in the inequality $n^{200}<5^{300}$?

\begin{aligned} &n^{200}<5^{300} \\ &\left(n^{2}\right)^{100}<\left(5^{3}\right)^{100} \\ &\left(n^{2}\right)^{100}<(125)^{100} \\ &n^{2}<125 \end{aligned}
Maximum value of $n$ is 11 .

OR

$200 \log n<300 \log 5$
\begin{aligned} &n<10^{\frac{3}{2} \log 5} \\ &n<11,18 \\ &\therefore n=11 \end{aligned}

OR

\begin{aligned}
&n^{200}<5^{300} \\
&\left(n^{2}\right)^{100}<\left(5^{3}\right)^{100} \\
&\sqrt{n^{2}}<\sqrt{5^{3}} \\
&n<5^{\frac{3}{2}} \\
&n<11,18 \\
&\therefore n=11
\end{aligned}

OR

\begin{aligned}
&n^{200}<5^{300} \\
&n<5^{\frac{300}{200}} \\
&n<11,18 \\
&\therefore n=11
\end{aligned}

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