How can log link still allow negative predicted values?

I have been attempting to get a meaningful outcome of my data for months, and could really do with some help where to go next. I am attempting to fit a model based on past cost data.
y = Total ($/ha/year) (all positive, min = 0.02, max = 2773.05)

x1 = Area (ha)
x2 = Continent (factor)
x3 = GDP ($/yr)
x3 Population density
x4 = Method (factor)
x5 = Type (factor)

In initial scatter graphs Total and Area were heavily skewed with a long right tail. I have so far attempted a linear regression with no logged variables, area logged, total logged and both logged, a GLM with log link, and a Gaussian GLM, and I consistently get negative predicted values. When I did log(Total+1) and log(Area+1) the fitted vs. residual graph appeared to have no negative fitted values but when I was using the predict function, one of my outputs was negative. Hat values seem high so perhaps some leverage is occurring. Does anyone have any ideas? Thanks.