Can I put Log(Y) as a dependent variable in a count data model I have count data passenger as Y. The data look like this, as many of the values are 1 (about 18%.)  
Does it make sense that I take a log of it, and take it as a dependent variable in a generalized linear model with Poisson distribution :  
I know the link function is log for Poisson distribution.  Did I have a problem to take double log of the Y? The question for me is that my Log(Y) model has a much better goodness-of-fit stat compared to my Y model. I tried some Poisson and Negative Binomial model and they are not fitting very well.  
What other strategies may I try to model this data?
 A: You data was zero-inflated (maybe more than 70% responses were zeros?). If both Poisson regression and negative binomial regression had bad fit, you should try Zero-inflated Poisson or even Zero-inflated negative binomial models. These mixture models have been proven to have better performance than using transformation. 
A: You can't apply a Poisson model to the variable called logP on your graph because it includes non-integers.  A Poisson model can only be used for integers.  You can probably still fit it in your software and get interpetable results, but you are not really using a Poisson model.
As @PeterFlom says, if your original variable is a count then log Y is not.  If the original variable is a count and a Poisson model does not fit, then try a negative binomial model before you give up and start transforming the variable.
A: You have given too little information to say much! Assuming you also have some (unstated) regressors $x$, you can use the old trick from before the time of GLMs, to use a usual linear regression after applying the variance-stabilizing transformation, which for the Poisson distribution is $\sqrt{Y}$. That is often a useful approach! 
But note that for count variables, a multiplicative model is often natural, and the usual Poisson (or negative binomial) regression  has a multiplicative expectation structure. But, if in your case an additive model is adequate, you can use the mentioned "trick".
