# What is the difference between log scale. and plotting logarithms?

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?

• If the input data is spaced far apart in magnitude, $x$ is replaced with $\log x$ to draw them in closer, the x-axis is changed to $\log x$. If the outputs are spaced far apart, $y=f(x)$ will usually be plotted as $Y=\log f(x)$. The x-axis remains unchanged – David Peterson Jan 22 '15 at 3:03

Suppose that $f(1) = 1, f(10) = 2, f(100) = 3.$ Let's suppose further that you measure position on your paper in centimeters, with the origin being at the origin of your graph.
If you plot $\log(x)$ vs $f(x)$, you'll plot points at $(0cm, 1cm), (1cm, 2cm),$ and $(2cm, 3cm)$.
If, on the other hand, you use the log paper's log-scale on the x-axis, let's suppose that the first "Decade" of the paper starts at the 0cm mark in the horizontal direction, and the leftmost vertical line of this decade is labelled "1", the second decade starts at the 1cm mark (and starts with "10"), and so on. Then for the point $f(10) = 2$, you'll go to $(1cm, 2cm)$; the other points you plot will be at $(0cm, 1cm)$ and $(2cm, 3cm)$.
The horizontal axis may be labelled $\log x$ in the sense that the physical distance from the vertical axis really is (up to a constant) the logarithm of the $x$-value that made you plot a point; the lines on the paper (and their labels "1", "10", "100") are just a way for you to easily exponentiate these distances to get the original value of $x$. In that sense, the paper is "taking the log" for you as you plot things: you have $x = 100$, you look for the vertical line labelled 100, and put a point there...and its distance from the $y$-axis turns out to be 2, which is $\log 100$.
Personally, I don't like it. I tend to label the thin vertical lines 1, 10, 100, etc., and label the axis $x$, but then I'm not an engineer or physicist. Maybe they know something I don't about graphs...