<s>[INST] Comparing two large numbers Can you compare two large exponential numbers, like $5^{44}$ and $4^{53}$ without taking their logs?
 [/INST] One approach is to figure that, roughly, $2^{10} \approx 10^3$, and $5^9 \approx 2,000,000 = 2 \cdot 10^6$. 
Then, 
\begin{align}
4^{53} &= (2^2)^{53} \\
&= 2^{106} \\
&= 2^{100} \cdot 2^6 \\
&= (2^{10})^{10} \cdot 2^6 \\
&\approx (10^3)^{10} \cdot 2^6 \\
&= 10^{30} \cdot 2^6
\end{align}
By way of comparison,
\begin{align}
5^{44} &= 5^{45} \div 5 \\
&= (5^9)^5 \div 5 \\
&\approx (2 \cdot 10^6)^5 \div 5 \\
&= 2^5 \cdot (10^6)^5 \div 5 \\
&= 10^{30} \cdot (2^5 \div 5).
\end{align}
Now, $2^6 > 2^5 \div 5$, so one might suppose $4^{53} > 5^{44}$.
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