Finding the inverse of a rational exponential function I am having trouble finding the inverse of the following function:
$$f(x)=\frac{5e^x}{9e^{x}-5}$$
I am able to get a fair ways through the problem and through the use of the rules of logarithms have reached this point:
$$\ln{y}=5x-\ln{(9e^{x} -5)}$$
I know that I need to get that last bit simplified, but I am not sure how to do it. If it were simply $\ln{9e^x}$, the problem would be trivial.
Any help as to how I can proceed is greatly appreciated.

$$y=f(x)=\frac{5e^x}{9e^x-5}$$
$$\implies e^x(9y)-5y=5e^x\implies e^x(9y-5)=5y\implies e^x=\frac{5y}{9y-5}$$
Applying logarithm wrt $x,$ $$x=\ln\left(\frac{5y}{9y-5}\right)$$
Now,  we can apply $\displaystyle\ln\left( \frac {a_1a_2\cdots}{b_1b_\cdots}\right)=\sum{\ln a_i}-\sum{\ln b_i}$
Clearly, for real $x,$  $\displaystyle\frac{5y}{9y-5}>0\iff y(9y-5)>0$
