MHB Prove $\pi^e > 5^{1.9}$ Without Calculator

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To prove that $\pi^e > 5^{1.9}$ without a calculator, one can use logarithmic properties and inequalities. By taking the natural logarithm of both sides, the comparison simplifies to $e \ln(\pi) > 1.9 \ln(5)$. Approximating values, $\ln(\pi) \approx 1.1447$ and $\ln(5) \approx 1.6094$, leads to the calculation of $e \ln(\pi) \approx 3.1139$ and $1.9 \ln(5) \approx 3.0569$. Since $3.1139 > 3.0569$, it follows that $\pi^e > 5^{1.9}$. The discussion humorously mentions the use of a slide rule, but emphasizes the need for a purely analytical approach.
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Prove, without the use of a calculator, $\pi^e>5^{1.9}$.
 
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Can I use my slide rule? (If I can find it. It's packed away.)

-Dan
 
Dan, if you use it in your solution, then I will have to report you to the authority and you will be fined $250. Hehehe...
 

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