Consider $X_1,X_2,…,X_n\sim N(\mu,\sigma^2)$.
The density of a single $X_i\sim N(\mu,\sigma^2)$ is $$f_{X_i}(x_i;\mu,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(\frac{−(x_i−\mu)^2}{2σ^2}\right).$$
How do I find the log-likelihood $\ln(L(\mu,\sigma^2|x_1,…,x_n))$?
$$\begin{align} L(\mu, \sigma^2)&=\prod_{i=1}^n f_{X_i}(x_i)\\ &=\prod_{i=1}^n\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(\frac{−(x_i−\mu)^2}{2σ^2}\right)\\ &=\frac{1}{(\sqrt{2\pi\sigma^2})^n}\exp\left(\sum_{i=1}^n\frac{−(x_i−\mu)^2}{2σ^2}\right)\\ \end{align} $$
So we have $$\begin{align} \ln L(\mu, \sigma^2)&=\underbrace{-n\ln(\sqrt{2\pi})}_{\text{constant}} - n\ln(\sqrt{\sigma^2}) + \sum_{i=1}^n\frac{−(x_i−\mu)^2}{2σ^2}\\ \end{align} $$