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\begin{problem}[6.6] But since $u$ and $v$ are harmonic, $u_{xx}+u_{yy}=0$ and $v_{xx}+v_{yy}=0$.\begin{problem}[6.6]  \textit{Show that, if $f$ is holomorphic and nonzero in $G$, then $ln|f(x,y)|$ is harmonic in G.}  \end{problem}  \begin{proof} 
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&v^{2}u^{2}_y+v^{2}uu_{yy}+v^{2}v^{2}_y+v^{3}v_{yy}-2u^{2}u^{2}_y-2uvv_yu_y-2uvu_yv_y-2v^{2}v^{2}_y\\  &-2u^{2}u^{2}_y-2uvv_yu_y-2uvu_yv_y-2v^{2}v^{2}_y  \end{align*}  But since $u$ and $v$ are harmonic, $u_{xx}+u_{yy}=0$ and $v_{xx}+v_{yy}=0$.  \end{proof}