I think that this example in particular is not the best for R^2. He's getting a really good fit for linear (especially when his first plot is centered in a narrow range), since log(x) has a nice Taylor expansions for log(x) ~ x - 1 in that region.
For fits that are almost entirely close to the mean (no slope) I would expect to be saved by the F-test, but we're not here since there's a region where a linear fit fits the data at least somewhat well.
For fits that are almost entirely close to the mean (no slope) I would expect to be saved by the F-test, but we're not here since there's a region where a linear fit fits the data at least somewhat well.