Solve the equation $(\ln x)^2 + \frac{2}{\log_x e} = 3$. Mrs Tan decides to track the relationship between the age, $x$ (in months) of her newborn baby girl and the circumference of her baby girl's head, $y$ (in centimetres).
After plotting the data collected for $1^{\text{st}}$, $6^{\text{th}}$, $12^{\text{th}}$, $18^{\text{th}}$ and $20^{\text{th}}$ month, the following graph was obtained.

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 $x = e^{-3}$