The image is correct, although the presentation is confusing because the association of the lines with the axes is not indicated clearly.
Plotting on an arithmetic scale (the "normal" sort of axis we all first learn about), an exponentially growing number of cells will result in a line that starts out looking pretty flat and then suddenly shoots up.
If you're interested in knowing the rate of growth, this sort of line isn't very useful, because the same rate will have different slopes, while the same slope indicates different rates, depending on the number.
If you instead plot the growth curve against a logarithmic axis, then you get a graph that focuses clearly on growth rate, with the same slope always indicating the same rate, no matter how many cells are present. Thus, if the growth rate is consistent, you'll see a straight line, as in the example presented.