Originally Posted by

**icemanfan** If you know how to do inverses, then you know that the first step is replacing all instances of x with y and all instances of y with x. I will run through this problem with you: $\displaystyle y = \frac{e^x}{1+2e^x}$

First, switch x and y: $\displaystyle x = \frac{e^y}{1 + 2e^y}$

Then multiply both sides by $\displaystyle 1 + 2e^y$: $\displaystyle x(1 + 2e^y) = e^y$

Distribute: $\displaystyle x + 2xe^y = e^y$

Subtract $\displaystyle 2xe^y$ from both sides: $\displaystyle x = e^y - 2xe^y$

Factor: $\displaystyle x = e^y(1 - 2x)$

Divide both sides by $\displaystyle 1 - 2x$: $\displaystyle \frac{x}{1-2x} = e^y$

Take the natural logarithm of both sides: $\displaystyle \ln{\frac{x}{1-2x}} = y$

And that's it.