<s>[INST] Number comparison: $5^{152}<2^{353}$ and $2^{1413}<3\cdot 5^{608}$ Is it possible to prove that $5^{152}<2^{353}$ and $2^{1413}<3\cdot 5^{608}$ without using a calculator or logarithms (middle school math only recommended)?
My idea for the first one was to use the obvious $5^3<2^7$ and then raise to the power of $50$ to get $5^{150}<2^{350}$. But since $5^2>2^3$, I couldn't use this approach to get the desired result.
Can you please help me find a relatively short proof for these inequalities? Thank you.
 [/INST] As @Gottfried Helms pointed out, the binomial expansion works.
We have
\begin{align*}
 &5^{152} < 2^{353} \\
 \iff \quad & 10^{152} < 2^{505}\\
 \iff \quad & 10^{152} < 1024^{50}\cdot 2^5\\
 \iff \quad & 10^{152} < 10^{150}(1 + 3/125)^{50}\cdot 2^5\\
 \iff \quad & \frac{25}{8} < (1 + 3/125)^{50} \\
 \iff \quad & \frac{5}{2\sqrt{2}} < (1 + 3/125)^{25} \\
 \Longleftarrow \quad & \frac{5}{2\sqrt{2}} < 1 + 25 \cdot \frac{3}{125}
 + \frac{25\cdot 24}{2}\frac{3^2}{125^2}\\
 \iff \quad & \frac{5}{2\sqrt{2}} < \frac{1108}{625}
\end{align*}
which is true.
We are done.
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