I'm trying to find the solutions of $\log(z)=i\log(\bar{z})$ where $\bar{z}$ is the conjugate of $z$. I'm aware of the multivalued complex log, so $\log(z)=\log|z|+i\arg(z)$ but I don't see to be getting anywhere,
any hints, hints only please, would be appreciated
Let's assume we've made a branch cut to make life easy on ourselves and the argument ranges from $0$ to $2\pi$. Then what we have is $z = |z|e^{i\theta}$ and $\bar{z} = |z|e^{-i\theta}$. Then, $\log(z) = \log|z|+i\theta$ and $\log(\bar{z}) = \log|z|-i\theta$. Making appropriate substitutions, we get
$$\log|z|+i\theta = i(\log|z|-i\theta) = i\log|z|+\theta.$$
Try equating the real and imaginary parts from here and you'll get a nice solution. (Just read your bit about hints only!)