I find plots in scientific literature beyond confusing. I understand quite clearly the difference between a linear and a logarithmic scale, and when each is desirable. Suppose we are plotting values for the equation $$ y = f(x)$$
If the points in $x$ vary over several orders of magnitude, a $\log$ scale is useful to capture all the points. However, quite often a log scale will be used, and the $x$ axis will be labeled as $\log(x)$, particularly in engineering literature.
Is this even correct, or just sloppy notation to indicate that a $\log$ is being used? For instance, if I read a value on the scale itself, is this not representing just $x$, instead of $\log(x)$?
Finally, how does plotting $\log(x)$, and keeping the scale linear, differ from plotting the raw value of $x$ on a logarithmic scale? In the first case, to understand the data visually, I have to somehow be able to exponentiate each $x$ value in my mind, while in the second case there is no work to be done.
Finally, a common phrase when using a $\log$ scale is "the paper is taking the log". What does this even mean?