How did the author got this final result from this Gaussian Distribution formula?
Take natural logarithm of the expression. Logarithm of product of two terms can be written as sum of logarithms of each term: log(a.b) = loga + logb. e and natural logarithm cancel each other, there is left first term in the result. Logarithm of a 1/number is equal to -log(number), it gives second term in the result.
Taking logarithms of exponentials is quite common method for preventing numerical overflows and ensuring numerical stability.