First, I'll identify key properties of the natural logarithm function:
- Domain: $(0, \infty)$, which restricts to $(0, 2]$ in our case
- The derivative is $f'(x) = \frac{1}{x}$, which is always positive for $x > 0$
- $\ln 1 = 0$
- $\ln x < 0$ for $0 < x < 1$
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