Answer
$y = \frac{a}{e^a - 1} (x - 1)$
Work Step by Step
The question asks for the equation of the line that passes through $(e^a, \ln (e^a)$ and the x-intercept of $\ln x$
Thus, at $x= e^a$, the y-coordinate would be equal to $\ln (e^a) = a$
The x-intercept of $\ln x$ is at (1, 0)
So the slope would be $\frac {a - 0} {e^a - 1}$ = $\frac{a}{e^a - 1}$
Thus we have $y = \frac{a}{e^a - 1} x + b$
Substitute (1,0) into the equation
$0 = \frac{a}{e^a - 1} (1) + b$
Thus $ b = -\frac{a}{e^a - 1}$
So $y = \frac{a}{e^a - 1} (x - 1)$