In continuous action spaces, how is the standard deviation, associated with Gaussian distribution from which actions are sampled, represented?

I have a question about implementing policy gradient methods for problems with continuous action spaces.

Assume that actions are sampled from a diagonal Gaussian distribution with mean vector $mu$ and standard deviation vector $sigma$. As far as I understand, we can define a neural network that takes the current state as the input and returns a $mu$ as its output. According to OpenAI Spinning Up, the standard deviation $sigma$ can be represented in two different ways:

I don’t completely understand the first method. Does it mean that we must set the log standard deviations to fix numbers? Then how do we choose these numbers?