1. ## Comples logarithms

Find an expression for z as a function of w in terms of the complex logarithm where

$\displaystyle w = f(z) := 2exp(z) + exp(2z)$

and z is complex.

I assume I need to get f(z) into a form that I can take logs, but I'm really struggling with getting started on this one.

I know that the answer is $\displaystyle z = log(-1 + exp(\frac{1}{2} log(1+w)))$.

Any help would be awsome.

2. ## Re: Comples logarithms

$\displaystyle e^{2z}= (e^z)^2$ so your equation can be written $\displaystyle w= 2e^z+ (e^z)^2$. Let $\displaystyle y= e^z$ and the equation becomes the quadratic equation $\displaystyle w= y^2+ 2y$ or $\displaystyle y^2+ 2y- w= 0$. Solve for y using the quadratic formula, then take the logarithm to solve $\displaystyle e^z= y$ for z.