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Thread: Comples logarithms

  1. #1
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    Comples logarithms

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

    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 z = log(-1 + exp(\frac{1}{2} log(1+w))).

    Any help would be awsome.
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  2. #2
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    Re: Comples logarithms

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