Check for endogeneity I run a log linear model, as $log(y) = b_0 + b_1x_1 + b_2x_2 + e$. I think $x_1$ may be endogenous and I would like to test it, so that I can consequently run a two-stage model. I would like to know if I there is a way to check it or if I must have at least one instrumental variable for $x_1$.
Another specification of my model is $log(y) = b_0 + b_1log(x_1) + b_2x_2 + e$. May I use the same procedure with a logarithm or is there a difference?
 A: In general, endogeneity is a theoretical property and not something that can be tested from the data at hand. Then you need something as an instrument, like you say. 
The second question sounds more like you are wondering what functional form will be best. There will certainly be a difference in the parameter values, but it may be that the predictions from the two are the same. You can run both, predict and inspect visually: 
You could for example estimate model 1 first and compute $\widehat{\log y_1}$ as the predicted values from the first model and $\widehat{\log y_2}$ as the predicted values from the second. Then you can plot them against each other.
Stata code could be 
reg logy x1 x2
predict yhat1 , xb
g logx1 = log(x1)
reg logy logx1 x2
predict yhat2 , xb 
twoway (scatter logy x1) (scatter yhat1 x1) (scatter yhat2 x1) , legend(order(1 "data" 2 "linear" 3 "logarithmic"))

A: This response takes issue with @superpronker's suggestion that endogeneity is not testable. To be specific, Wooldridge offers an explicit test for it in the first edition of his book, Econometric Analysis of Cross-Section and Panel Data, chap. 6.2.1, beginning on p. 118, where he writes:

We start with the linear model and a single possibly endogenous
  variable. For notational clarity we now denote the dependent variable
  by y1 and the potentially endogenous explanatory variable by
  y2...The population model is
y1=z1 d1 + a1 y2 + u1 
The maintained exogeneity assumption is
E(z' u1)=0
(Following a suggestion by Hausman) a formal test of endogeneity is:
  if y2 is uncorrelated with u1, the OLS and 2SLS estimators should
  differ only by sampling error.

Basically, Wooldridge is saying that if the residuals are significantly correlated with the explanatory variable(s) then there is evidence for endogeneity and "2SLS is probably a good idea," (p. 120)...
