Your line is being plotted, you just can't see it in the window because the values are quite different. What is happening when you include the log='xy'
argument is that the space underneath the plot (so to speak) is being distorted (stretched and/or compressed), nonetheless, the original numbers are still being used. (Imagine you are plotting these points by hand on graph paper; you are still marking a point where the faint blue graph lines for, say, (1,500) cross, but the graph paper has been continuously stretched such that the lines are not equally spaced anymore.) On the other hand, your model is using the transformed data.
You need to make your plot with the same transformed data as your model, and then simply re-mark your axes in a way that will be sufficiently intuitively accessible. This is a first try:
plot(log(x), log(y), axes=FALSE, xlab="X", ylab="Y")
box()
axis(side=1, at=log(c(1,2, 10,20, 100,200)),
labels=c( 1,2, 10,20, 100,200))
axis(side=2, at=log(c(125,135, 250,260, 350, 500)),
labels=c( 125,135, 250,260, 350, 500))
abline(model, col="red")