Question

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

Answer 1

\[
\begin{aligned}
&n^{200}<5^{300} \\
&\left(n^{2}\right)^{100}<\left(5^{3}\right)^{100} \\
&\left(n^{2}\right)^{100}<(125)^{100} \\
&n^{2}<125
\end{aligned}
\]
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}