If I have log returns for a specific stock, then the weekly log return is the log of Friday's closing price minus the log of Monday's closing price, i.e. $R_{weekly} = log(Price_{Friday}) - log(Price_{Monday})$.
However, can I also calculate the log return as the sum of the daily log returns? $R_{weekly}=[log(Price_{Monday}) - log(Price_{Sunday})] + [log(Price_{Tuesday}) - log(Price_{Monday})] + {...} + [log(Price_{Friday}) - log(Price_{Thursday})]$
Your second formula regarding the sum of day-to-day returns collapses as follows:
$$ \begin{align} R_{weekly,2} &= \text{log}(Price_{Mon}) - \text{log}(Price_{Sun}) \\ &+ \text{log}(Price_{Tue}) - \text{log}(Price_{Mon}) \\ &+ \dotsc \\ &+ \text{log}(Price_{Fri}) - \text{log}(Price_{Thu}) \\ &= \text{log}(Price_{Fri}) - \text{log}(Price_{Sun}) \end{align} $$
Compared to the first formula,
$$ R_{weekly,1} = \text{log}(Price_{Fri}) - \text{log}(Price_{Mon}), $$
the difference is
$$ R_{weekly,2} - R_{weekly,1} = \text{log}(Price_{Mon}) - \text{log}(Price_{Sun}). $$
That is, you include one extra day-to-day return in the definition of $R_{weekly,2}$ as compared to the definition of $R_{weekly,1}$.
In my understanding, the relevant weekly return is the cumulative return over all seven days of the week (if there are no trades on, say, Sunday, then define the corresponding day-to-day return as zero), which collapses to
$$ R_{weekly} = \text{log}(Price_{Fri,\ this \ week}) - \text{log}(Price_{Fri, \ last \ week}) $$
(which is what @SimoneBortolato suggested before). However, depending on what you want to do with those weekly returns, other definitions could make sense as well.
I can't still comment a question, since I've still got low reputation, so I'm going to answer. I think you should calculate weekly returns from friday to friday (close of the week to close of the following week).
As mentioned above, $\log(p_{\text{next Friday}})-\log(p_{\text{this Firday}})$ is correct for weekly calculations of the logarithmic transformation of returns.
There are three reasons to use the logarithmic transformation of returns.
First, custom says to use it. The tradition was developed because if you used a slide rule or a punchcard computing device, it was easier and faster. The one slight loss from removing slide rules from general use is a reduction in numeracy. The slide rule gives you a different feel for the relationships among numbers.
Of course, if you want to do mass computing with something other than a mechanical or cathode ray-tube computer and you do not want to learn to use a slide rule for the same reason you also probably could not use a sextant, then the appeal to tradition or convention is probably not well-founded.
The second reason is to linearize the data. Usually, linearization is more to support the researcher's cognitive processes than serve a mathematical purpose. It can be easier to see structural breaks and things like that because it appears as a bend in a line. The exception to that rule is for models constructed in a logarithmic space.
Models built in a logarithmic space should be solved in a logarithmic space.
The third reason to use a logarithmic transformation in models of capital is to assure the existence of a first moment. Unfortunately, the distribution is unlikely to have a defined covariance matrix, and as such, it is difficult to discuss what least-squares models are really doing. The use of a Bayesian model would help in that case.
There is nothing wrong with using log-returns; in fact, it is a good decision in some models because of how they are built. I thought I would add this bit of warning, with the intent of being helpful, so that you would stop for a second and be sure that you are doing what you are doing for a good reason and not because others have done the same thing.
I have a friend who is an engineer. Growing up, she watched her mother make meatloaf. When she got her own apartment, she invited her family and friends over for dinner. She made meatloaf.
She had thought she had memorized her mother's recipe. She put it in the oven and baked it with all of the ingredients that she could remember. Her family arrived a little early, and her mom came in and helped her in the kitchen.
She pulled the meatloaf out of the oven and let it cool a little. She then took it out of the pan, cut off the ends, and put it on a dish to serve.
Her mother asked her why she had cut off the ends.
She said, "you always cut off the ends of the meatloaf."
Her mother said, "that is because my serving dish is smaller than the loaf pan I bake it in. If I didn't cut off the ends, it wouldn't fit. It would fall over the sides. Your serving dish is large enough; you should not cut the ends off."
Just be sure that you are doing it for reasons two or three and not for reason one. Of course, you are still just learning. It is good that you asked questions here. In addition to how, please feel free to add who, what, where, when, and why to your questions. We might be able to give you more targeted advice.
Friday to Friday is correct (close of the week to close of the following week)
Suppose the following closing prices
Then, Friday's close profit is \$100 and the return is $\frac{\\\$100}{\\\$1000}=10\%$.
