I think you and the author are using terminology differently. That graph is absolutely a logarithmic axis to me. The five ticks on the axis are equidistant but each represents a number 10x of the previous. That's nonlinear to me. My definition of linear growth is that it is bounded above by a linear function. Its first derivative would therefore be bounded above by a constant.

If something is a straight line when you plot it in log scale, you are plotting exponential growth.