I have a question regarding the the contracted Christoffel symbols from David Tongs PDF on general relativity.
He wants to prove that $$\Gamma^{\mu}_{\mu v}=\frac{1}{\sqrt{g}}\partial_v\sqrt{g}$$ Where $g=\text{det }g_{\mu v}$, is the determinant of the metric.
In one step of the proof he writes $$\Gamma^{\mu}_{\mu v}=\frac{1}{2}g^{\mu\rho}\partial_vg_{\mu\rho}=\frac{1}{2}tr(g^{-1}\partial_vg)=\frac{1}{2}tr(\partial_v\text{log }g)$$
I cant really understand what is going on here, could someone explain the two steps in this line?
