Find $z$ as a function of $w$ in terms of the complex logarithm, where $w=f(z):=2e^z+e^{2z}$

Question

I have solved the following problem but would like to double check that I did it properly.

The problem says: Find an expression for $z$ as a function of $w$ in terms of the complex logarithm, where $w=f(z):=2e^z+e^{2z}$, and use it to find all values of $z$ for which $f(z)=3$.

So I have the following:

We have $w=2e^z+e^{2z}$, so taking complex logarithms on both sides we get $$\log(w)=\log(2e^z+e^{2z})=\log(e^z(2+e^z))=\log(e^z)+\log(2+e^z)=z+\log(2+e^z)$$

And I am kind of stuck here because I don't know how to deal with $\log(2+e^z)$.

Any help would be very much appreciated... I know once I have the expression, all I hav to do is substitute $w=3$ into the equation, but I cannot get to that equation.

Thanks!

Answer 1

Hint: Let $y=e^z$ and solve the quadratic equation

$$ w= 2y+y^2 $$

See a related problem.