Take the $R^2$s over log-linear and log-log transformed data with a *massive* grain of salt. The usual tools for linear regression assume that any noise in your data is (1) gaussian and (2) uniform over the data set. However, a linear-to-log transformation will magnify the noise in places where the signal is smaller. (To see why, plot the error bars on the log scale. What happens to the error bar of $\log(x)$ if $\Delta x$ is bigger than $|x|$?) There are specific ways to deal with this in the statistics (via nonlinear model fits, or non-uniform noise profiles)... —
Emilio Pisanty 1 min ago