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Which is larger: 4^(6^5) or 5^(3^8)? Question
This is a [calculation-puzzle], which is a puzzle that involves numerical calculations, generally using the basic operations: addition, subtraction, multiplication, and division.
Without using a computer or calculator, determine which number is larger:$$\large4^{6^5} \text{ or }5^{3^8}$$
Rules:
- You cannot use a computer or calculator
- You can use logarithms. To reduce some arithmetic, a limited table of values are supplied (depending on what base log you prefer):$$\begin{array}{|r|c|c|}\hline&\ln x\text{ (base }e\text{)}&\log x\text{ (base }10\text{)}\\\hline2&0.6931&0.3010\\\hline3&1.0986&0.4471\\\hline5&1.6094&0.6989\\\hline\end{array}$$For some additional ease, you can use $\ln(1+x)\approx x-\frac{x^2}2+\frac{x^3}3$ for small $x$ if you need it.
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