Finding the Inverse of a function with natural logs

Question

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.

Answer 1

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 $$f^{-1}(x) = \ln\left( \frac{6 x + 2}{5x - 7} \right).$$