Why do people use log(x+1) or log(y+1) as their independent/dependent variable? Why do people use log(x+1) or log(y+1) as their independent/dependent variable? In these cases, is it that different from using just log(x) or log(y)? What is the benefit of using log(x+1) or log(y+1)?
Sometimes when I read papers, the authors specify that their model's variable is the log of a variable + 1. I have been trying to read through the posts on why this is the case, but still having a hard time figuring it out.. can someone explain to me please?

The answer is that people do it when they have zero-values in their data. When that happens, you cannot log your data, since $\log(0)$ is undefined. However, if you add some small value, such as $1$, then you can take the logarithm of that small value.
This practice, however, is of questionable statistical validity.
