Norwich (2003) gives the following equality based on a substitution as shown in the image.
I don't get it though because $\frac{1}{2}$ appears in two places in the first formula. Does the substitution written occur at both places? Even if I do this, I don't know how to distribute the multiplication of "$\ln e$"
Could someone show the steps in-between involving the substitution, and how to arrive at the final equation shown.
We have $$ \frac12\ln(2\pi\sigma^2) + \frac12 = \frac12\ln(2\pi\sigma^2) + \frac12\ln e\\ = \frac12\left(\ln(2\pi\sigma^2) + \ln e\right)\\ = \frac12\ln(2\pi e\sigma^2) $$ It is the ${}+\frac12$ on the right side that gets an extra factor of $\ln e$. Then we use distribution, and finally elementary logarithm rules to put the two logarithms together into one.