I have read other links on here about when to take the logarithm of a dependent (or independent) variable. I understand that the log transformation gives a % interpretation rather than a levels interpretation, which might be more desirable because a 1 unit increase means different things depending on the initial starting point. I also understand that if a variable is log-normally distributed like income, then taking the log makes sense. Also, if the theory says log is appropriate, sure.
But, what about variables like population density in a region or the child-teacher ratio for each school district or the number of homicides per 1000 in the population. I have seen professors take the log of these variables. It is not clear to me why. For example, isn't the homicide rate already a percentage? The log would the the percentage change of the rate? Why would the log of child-teacher ratio be preferred? Suppose there are no theoretical models supporting that the relationships should be log.
Is it the case that we view a class-size of say 20 students so differently from say 25 students that a 1-student increase in class sizes means entirely different things depending on if we are 20 or 25? The same with the homicide rate? A homicide rate of 1 murders per 1000 means something so different from 10 per 1000 that an increase of 1 has very different meanings?
