<s>[INST] Finding the Inverse of a function with natural logs The function is 
$$f(x)= \frac{7 e^{x} +2}{5 e^{x} - 6}$$
 and I am suppose to find the inverse, so I switched the $x$'s and $y$'s.
I know I am suppose to multiple the denominator out and I end up with $5 \, e^{y} \, x - 6 x = 7 \, e^{y} + 2$. From there I do not know what to do.  
 [/INST] Consider:
$$f(x) = \frac{7 \, e^{x} + 2}{5 \, e^{x} - 6}$$
for which
\begin{align}
y &= \frac{7 \, e^{x} + 2}{5 \, e^{x} - 6} \\
5 y \, e^{x} - 6 y &= 7 \, e^{x} + 2 \\
(5 y - 7) \, e^{x} &= 6 y + 2 \\
e^{x} &= \frac{6 y + 2}{5y - 7} \\
x &= \ln\left( \frac{6 y + 2}{5y - 7} \right)
\end{align}
It can now be stated that
$$f^{-1}(x) = \ln\left( \frac{6 x + 2}{5x - 7} \right).$$
</s>