How can I isolate for the $z$ exponent?

Question

Can anyone help me with this math equation?

Solve for $z$

$$P = \frac{e^z}{1 + e^z}$$

$$P(1 + e^z) = e^z$$

$$P + Pe^z = e^z$$

$$P = e^z - Pe^z$$

I've got this far, am I at least on the right track? Not sure exactly what to do next.

Answer 1

Hint: $P= \frac{e^z+1-1}{e^z+1}=1-\frac{1}{e^z+1} $

Answer 2

You are on the right track. Now that you are at

$$P = e^z -Pe^z$$

Factor out $e^z$, that is

$$P = e^z(1-P),$$ and dividing yields

$$\frac{P}{1-P}=e^z.$$

We finish off by using the logarithm

$$z = \ln\left(\frac{P}{1-P}\right). $$