Starting from the log-likelihood function for a Gaussian,
$$ \mathcal{L}\mathcal{L} = - \frac{N}{2} \log{\left(2 \pi\sigma^2\right)} - \frac{1}{2 \sigma^2} \sum_{i=1}^{N}{{\left(x_i - \mu\right)}^2} \tag{1} $$
if we assume the population to be well-represented by the sample distribution such that $ \sigma^2 = \frac{\sum_{i=1}^{N}{{\left(x_i - \mu\right)}^2}}{N} $ , can we simplify the term on the right to $\frac{N}{2}$, resulting in:
$$ \mathcal{L}\mathcal{L} = -\frac{N}{2} \left(\log{\left(2 \pi\sigma^2\right)} + 1 \right) \tag{2} $$