log rules for MLE

Question

Does anyone know how I get from,

$$\ln\left\{\frac{1}{\sigma}\frac{1}{\sqrt{2\pi}}\right\}$$

To

$$-\frac{1}{2}\ln{\sigma^2-}\frac{1}{2}\ln2\pi$$

Answer 1

Continuing from you, \begin{align} -\ln(\sigma)-\frac12\ln(2\pi)&=-\ln\left((\sigma^2)^{\frac12}\right)-\frac12\ln(2\pi)\\ &=-\frac12\ln\sigma^2-\frac12\ln2\pi \end{align}