Question about Bernoulli Distribution calculation

Question

can sombody explain the above calculation in the red circle marked with "why?"? I am studying MLE with Bernoulli Distribution, and in the middle of a video clip, the lecturer says $ 1\over{n} \sum_{i=1}^{n}x_i$ becomes $ n\bar x $. But I can't see anything like $1\over{n}$ in any of the calculation steps in the picture.

Answer 1

Equations shown involve straightforward application of rules of logarithms. In the red circle, all he is saying is that the sum_i(x_i) -- showing at the bottom in the product log(p)*sum_i(x_i) -- can alternatively be written as N*average(x_i).

That's it.

Answer 2

Logarithms are not needed here. I think it's pretty clear $$\frac{1}{N}\sum^N_{i=1}x_i = \frac{x_1 + x_2 + ... + x_N}{N} = \overline{x}$$ by the definition of what a mean is ($\overline{x}$ represents the mean of $x$).