Log and graphs

Rearranging, $\ln^2(y) = \ln^2(x) - \cfrac{1}{e} \geq 0$ , so $\ln^2(x) \geq \cfrac{1}{e}$ , which means $\ln(x) \leq -\cfrac{1}{\sqrt{e}}$ and $\ln(x) \geq \cfrac{1}{\sqrt{e}}$ , or in other words, $x \leq e^{-\frac{1}{\sqrt{e}}}$ and $x \geq e^{\frac{1}{\sqrt{e}}}$ , so the $x$ -coordinates of the two colored points are $x_1 = e^{-\frac{1}{\sqrt{e}}}$ and $x_2 = e^{\frac{1}{\sqrt{e}}}$ .

Substituting either of these $x$ values back into $\ln^2(y) = \ln^2(x) - \cfrac{1}{e}$ solves to $y = 1$ , so the distance between the two colored points is $e^{\frac{1}{\sqrt{e}}} - e^{-\frac{1}{\sqrt{e}}}$ .

Therefore, $a = \cfrac{1}{\sqrt{e}}$ , and $\ln a = -\cfrac{1}{2} = \boxed{-0.5}$ .