How to Take Log of Zero Income 

 I encounter a problem when using a Log normal distribution to model income distribution. Namely, there are a bunch of people in my dataset who report zero income, maybe due to unemployment, and I am wondering how to logarize the zero incomes. I notice some researchers just drop the observations with zero income while others assign a small amount of income to them so that logarithm can be taken legitimately. Obviously, we can try both ways to see how the results stand. But I am wondering if there are some experts on this topic who can clarify the pros and cons of these and other approaches treating zero incomes.  

 A related question is what model you think fits the income distribution best, a Lognormal, a power distribution, or a mixture model of a Normal and a point mass at zero, and so on.  
Look forward to your thoughts on these questions. 

 Lastly,  here  is an interesting animation of the income distribution in the USA.