The author has no clue of what is doing. Like zero.

He doesn't understand what the model is trying to do. We are showing that as the system grows in orders of magnitude of 10s, 100s, 1000s, the system takes equivalent times that are also scales of 10s, 100s, 1000s. That is what is shown the graph. Is that true or not? YES. It is evident to the f. eyes. If your math shows it is not true then you did your math wrong or you misused it. This is why physicists are better at this sort of no-nonsense data analysis. There is nothing illogical about using time in the x-axis if the question is about growth in time. It is exactly the quantity you want to understand and that drives the system. Now of course it could be and should be some other mechanism (also a function of time) that is the underlying cause (or causes) but time is what we can measure easily and plot on the graph. There is a clear linear relationship (on average) between the log of time and the log of price. Again, look at the damn graph.