Given that the model distributon is Gaussian with known variance $\sigma^2 = 1$
$$ p(x_i | \mu, \sigma^2=1) = \frac{1}{\sqrt{2\pi}}\exp{-\frac{(x_i - \mu)^2}{2}} ,$$
the log of the data likelihood $\mathcal{D} = \{x_0,x_1,\dots, x_n \}$ can be expressed as function of $\mu$ as following: $$ \ln{p(D|\mu)} = -\frac{1}{2}\Big(- 2\sum_{i=1}^n x_i\mu + n\mu^2 \Big) + \text{const}$$
where $\text{const}$ utilizes all of the term which do not depend on $\mu$.
Further, assume the prior distribution over mean value ${\mu}$ is Gaussian with ${\mu_0}$ and variance ${\sigma_0^2 = \frac{1}{\tau_0}}$, then I have shown already that the log posterior is given by
$$\ln p(\mu |X, \mu_0, \tau_0)= -\frac{1}{2}\Bigl( (n + \tau_0)\mu^2 -2(\sum_{i=1}^n x_i + \tau_0\mu_0)\mu\Bigr) + \text{const}.$$
How can I evaluate the posterior in the log space and show that it is Gaussian with parameters $\hat \mu, \hat \sigma^2$? Just multiplying the prior with the likelihood gives me a nasty term where I don't really see what my $\bar \mu, \bar \sigma^2$ could be, e.g. I don't know how tog et it into a "Gaussian expression".