When to use Log in Regression? I saw this sentence:
"I use log(income) partly because of skewness in this variable but also because income is better considered on a multiplicative rather than additive scale. 
In other words, \$1,000 is worth a lot more to a poor person than a millionaire because \$1,000 is a much greater fraction of the poor person’s wealth"
on page 143 on this link http://www.biostat.jhsph.edu/~iruczins/teaching/jf/ch12.pdf.
But when I check their skewness using library(e1071) in R (as seen below), I found out that the skewness of income is not that high or low. My question is how do I determine if I need to used log transformation in a regression model?
PS the chicago data is in library(faraway)
> skewness(chicago$race)
  [1] 0.5570103
> skewness(chicago$race)
  [1] 0.5570103
> skewness(chicago$fire)
  [1] 1.271188
> skewness(chicago$theft)
  [1] 2.955751
> skewness(chicago$age)
  [1] -0.9210877
> skewness(chicago$income)
  [1] 1.155
> skewness(chicago$involact)
  [1] 0.8079598

 A: Income is commonly accepted be right skewed, where people making disproportionately large amounts of income pull the mean much higher than the median. Its skewness in that particular data set may not contradict the prevailing norm.
Using log income also lowers the impact of heteroskedasticity. However this is not the best use of it, if heteroskedasticity is a problem you may want to use GLS.
Your primary question: You use log transform for the reason mentioned above, if you believe the increase to be relevant proportionally (+1% income) rather than linearly (+1$ income). This is a question of your theory or functional form. Is a dollar the same for a millionare and for a pauper? Choose linear in this case. If a dollar does nothing for a millionaire but a lot for a pauper, choose ln(). 
If your dependent variable is also in logs, your coefficient is an elasticity, a very important concept in economics.  A coefficient is still called a semielasticity even if the independent variable is in logs and the dependent variable is not! 
