# How do you solve 40 / (1+e^(-2x)) = 8?

Jul 7, 2015

When solving a complicated equation, try and undo all the things being done to the variable, and in the reverse order!

#### Explanation:

In this case, multiply through by the denominator and get

$40 = 8 \cdot \left(1 + {e}^{- 2 x}\right)$

Now isolate the $x ,$ step-by-step:

$5 = 1 + {e}^{- 2 x}$

$4 = {e}^{- 2 x}$

$4 = \frac{1}{{e}^{2 x}} \iff {e}^{2 x} = \frac{1}{4}$

This means that you get

$2 x = \ln \left(\frac{1}{4}\right) = - \ln 4$ ;

$x = - \ln 2$ , or $x \cong - 0.693$.

I left out a couple of steps near the end to make you think!