You can use log or simple returns, depends on your data set. Will get to that in a bit.
How can you calculate the SD with just one closing price and the mean? You need a time series of data to calculate the SD.
Like I said, you can take log of simple returns. This depends on the data set. If you are dealing with multiple years data, then simple is better, else stick to log. Log returns tend to give you a fairly representation of returns.
Let me give you an example. You invest in a stock -
Year 1 - 100
Year 2 - 150
Year 3 - 100
So 2nd year, you make a 50% return (simple), and year 3 you get a -33% return. If you take the average across 2 years, then it means you’ve earned 8.33% return. But in reality you started with 100 and ended up with 100.