Three Solutions Exactly

Let $f(x) = \left(\mathrm{e}^{ax} - 1\right)\ln x$, where $a > 0$.

1. When $a = 1$, find the area of the triangle formed by the two coordinate axes and the tangent line to the curve $y = f(x)$ at $(1, f(1))$. 2. If the equation $f(x) = ax^2 - ax$ has exactly $3$ real solutions on $[1, +\infty)$, find the range of possible values of $a$.