I am confused as to how the inverse of $f(x)=\frac{3+\ln x}{3-\ln x}$ is found. The problem within my textbook is $f(x)=$$\frac{3+\ln{x}}{3-\ln{x}}$ I checked the answer and it is $y=$$e^\frac{3x-3}{x+1}$
I've tried many times to simplify the equation after switching the variables but I don't know how to separate the $y$ from the natural logarithm. If someone could show me the steps involved I would appreciate it.

Steps:
Start with:$$y=\frac{3+\ln x}{3-\ln x}$$
Rewrite the equality as: $$\ln x=g(y)$$ and conclude that: $$x=e^{\ln x}=e^{g(y)}$$
Now switch $x$ and $y$ to get the final result.
