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Is the logarithmic return commonly used as QuantConnect states?

I am learning trading algorithm on QuantConnect, and I was confused when I read the rate of return part in the tutorial of introduction to financial python. See the original article by Tutorials - Introduction to Financial Python - Rate of Return, Mean and Variance - QuantConnect.com

It states that the logarithmic return is frequently used when calculating returns, and monthly return can be calculated by simply summing the daily returns up.

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There are some typos in the article I think, but let's look at the statement itself. When we are holding equity such as the Apple stock used as an example in the article, I suppose the return should be holding period return since there is no interest generated, am I right? Why a continuously compounded return can be applied here?

In addition, is the below reasoning correct? It links the two equations together to get the logarithmic return, but the two "r"s actually stand for different meanings. (see the remark I put)

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I am applying to the MFE programs this year, and I am preparing myself by learning on QuantConnect. So I am not pretty sure whether this is practically used in the industry, and I posting here to seek clarification from you. Thanks.
 
For your question about why are they are using a continuous compounded interest; because it create a continuous function across time allowing for easier application of calculus.
 
I got your point. In this case, since the calculation results are different using the two methods, how is this difference manifested in practice? By adding a footnote or something else?

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