First, our desired scales are in log-scale. This is not the default, so this change needs to be added through a scales layer. A quick look at the cheat sheet reveals the scale_x_continuous function lets us control the behavior of scales. We use them like this:

p + geom_point(size = 3) +  
  geom_text(nudge_x = 0.05) + 
  scale_x_continuous(trans = "log10") +
  scale_y_continuous(trans = "log10") 

Because we are in the log-scale now, the nudge must be made smaller.

This particular transformation is so common that ggplot2 provides the specialized functions scale_x_log10 and scale_y_log10, which we can use to rewrite the code like this:

p + geom_point(size = 3) +  
  geom_text(nudge_x = 0.05) + 
  scale_x_log10() +
  scale_y_log10()