# Comparing two numbers without using a calculator

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!

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

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 $$3/2>\ln 3$$

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

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

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