解方程 $(\ln x)^2 + \frac{2}{\log_x e} = 3$。 陈太太决定追踪她新生女婴的年龄 $x$（以月为单位）与女婴头围 $y$（以厘米为单位）之间的关系。
绘制了第 $1$、第 $6$、第 $12$、第 $18$ 和第 $20$ 个月收集的数据后，得到了以下图表。

1. Apply the change of base formula to express $\log_x e$ in terms of natural logarithms.

$\log_x e = \frac{\ln e}{\ln x} = \frac{1}{\ln x}$

2. Substitute the expression back into the equation to form a quadratic equation in $\ln x$.

$(\ln x)^2 + \frac{2}{\frac{1}{\ln x}} = 3$
$(\ln x)^2 + 2\ln x - 3 = 0$

3. Solve the quadratic equation for $\ln x$.

Let $u = \ln x$.
$u^2 + 2u - 3 = 0$
$(u - 1)(u + 3) = 0$
$u = 1$ or $u = -3$
$\ln x = 1$ or $\ln x = -3$

4. Solve for $x$.

$x = e^1 = e$ or $x = e^{-3}$