In the complex field, is this function Harmonic?

[oab729]

Homework Statement

Without directly verifying via the laplace equations, explain why Log|z| is harmonic in the punctured complex plane.

Homework Equations

The Attempt at a Solution

I thought it was because Log z is analytic on the complex plane except for the nonpositive real axis, so Log |z| would be analytic and hence harmonic since any |z| turns z into a positive real (for z=/= 0), hence Log |z| is like Log z for positive reals. But, if it's that, why would they ask to show its harmonic and not analytic.

 

[Billy Bob]

Do you have the theorem that if f=u+iv is analytic then u is harmonic?