a.) Show that if $v(x,y)$ is a harmonic conjugate of $u(x,y)$ in a domain $D$, then every harmonic conjugate of $u(x,y)$ in $D$ must be of the form $v(x,y)+a$, where $a$ is a real constant.
b.) Suppose that $f(z)$ is analytic and nonzero in a domain $D$. Prove that $ln|f(z)|$ is harmonic in $D$.
I know initutively why a is true, I also know that I must use Cauchy-Riemann equations to prove it but I do not know how to apply it correctly. For b, I am stumped.