How do I find the harmonic conjugate of a composite of two functions

Question

Given an entire function $f(z)$ how do I find the harmonic conjugate of $ log(|f'(z)|) $.

I tried directly solving the differential equations obtained via Cauchy–Riemann equations as follows:

set $f(z) = u(x, y) + iv(x, y)$ and let $$\psi(x, y) = log(|f'(z)|) = \frac{1}{2}log(u_x^2 + v_x^2) = \frac{1}{2}log(u_y^2 + v_y^2)$$ and hence, $$ \psi_x = \frac{u_yu_{xy} + v_yv_{xy}}{u_y^2 + v_y^2} $$ and $$\psi_y = \frac{u_xu_{xy} + v_xv_{xy}}{u_x^2 + v_x^2}$$

I tried to further use that $ \psi_x = \phi_y $ and $ \phi_x = -\psi_y $ to solve for $ \phi(x, y) $ however that was useless. How else should I proceed with finding the harmonic conjugate?

Answer 1

$\log g=\log |g|+i\arg g$ whenever RHS makes sense. (since $arg$ is multivalued, RHS may make sense only locally, so only near any point where $g \ne 0$, and it may be stitched together under certain circumstances like simple connectedness of the domain $U$ where $g$ doesn't vanish, $\Re g >0$ or even better but harder to check, $g+ c \ne 0, c \ge 0$ and the like)

So the conjugate harmonic of $\log|f'|$ is $\arg f'$ with the comments above that it may make sense only locally but not globally