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I need to compare $\frac{3}{2}$ and $\ln 3$ (which is the same as $\log_e{3}$). But I need to do it without any computer etc. Thanks in advance!

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Hint: $e* \sqrt e > 2.5 *1.5 >3 $

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Exponentiate both and compare the results.

Note that $$e^x=1+x+x^2/2 +...$$

Therefore $$e^{3/2} =1+3/2 +9/8+...>3$$

Thus $$3/2>\ln 3$$

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This is the same as comparing $e^{1.5}$ and $3$ by exponentiating both terms. But $$e^{1.5}=e\cdot e^{0.5} \gt e\cdot 2^{0.5} \gt 2.7\cdot 1.4 \gt 3$$

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Hint: compare $e^{1.5}$ and $3$. Should be able to do it knowing that $e \approx 2.7$

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Use the beginning of Taylor's expansion for $\mathrm e^{\tfrac32}$: $$\mathrm e^{\tfrac32}>1+\dfrac32+\dfrac12\,\dfrac94=\dfrac{29}8>3.$$

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