How do you solve an equation when you have both y and ln(y)?

Question

I'm trying to solve the equation $x = 8 + y + \ln (y-3)$ for $y$, but am having trouble isolating $y$. How can I get the $y$ out of the ln function without making the other $y$ the exponent of $e$?

Answer 1

You have to use Lambert function W:

$x-11=y-3 + \ln(y-3)$

$e^{x-11}=(y-3)e^{y-3}$

$y-3=W(e^{x-11})$