Then further note that
$$\mathrm{det} (g_{\mu \nu})=g=\frac{1}{4!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_1 \nu_1} \cdots g_{\mu_4 \nu_4}.$$
Here and in the following I use the notation
$$\Delta^{\mu_1 \ldots \mu_4}=\Delta_{\mu_1 \ldots \mu_r}$$
for the Levi-Civita symbols (which are NOT tensor components), i.e., the totally antisymmetric symbols with ##\Delta^{0123}=\Delta_{0123}=+1##.
Now take the derivative ##\partial_{\mu}## which gives
$$\partial_{\mu} g = \frac{1}{3!} (\partial_{\mu} g_{\mu_1 \nu_1}) \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_2 \nu_2} \cdots g_{\mu_4 \nu_4}=:\tilde{g}^{\mu_1 \nu_1} \partial_{\mu} g_{\mu_1 \nu_1}.$$
But now let's think about
$$\tilde{g}^{\mu_1 \nu_1} = \frac{1}{3!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_2 \ldots \nu_4} g_{\mu_2 \nu_1} \cdots g_{\mu_4 \nu_4}$$
is. To that end we evaluate
$$g_{\alpha \mu_1} \tilde{g}^{\mu_1 \nu_1}= \frac{1}{3!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_1 \nu_1} \cdots g_{\mu_4 \nu_4} g_{\mu_1 \alpha}.$$
In the last step I used the symmetry of the metric, ##g_{\mu_1 \alpha}=g_{\alpha \mu_1}##. Now do the contraction wrt. the ##\mu_k##, which leads to
$$g_{\alpha \mu_1} \tilde{g}^{\mu_1 \nu_1}=g \frac{1}{3!} \Delta_{\alpha \nu_2\cdots \nu_1} \Delta ^{\nu_1 \ldots \nu_4} = g \delta^{\nu_1}_{\alpha}.$$
From this you have
$$\tilde{g}^{\mu_1 \nu_1}=g g^{\mu_1 \nu_1}.$$
So you get
$$\partial_{\mu} g=g g^{\mu_1 \nu_1} \partial_{\mu} g_{\mu_1 \nu_1}.$$
So finally
$$\partial_{\mu} \ln (\sqrt{-g})=\frac{1}{2} \partial_\mu \ln(-g)=\frac{1}{2g} \partial_{\mu} g=\frac{1}{2} g^{\alpha \beta} \partial_{\mu} g_{\alpha \beta},$$
which is what you wanted to prove.