You can plot with exponential scaling, for example $x = 10^{x'}$. This is equivalent to plotting $\log_b f(x)$ vs. $x$.
To elaborate, consider a plot $P$ to be defined by plot coordinates $x', y'$ and a graph with points $x,y$.
If you want $x'$ to be "shorter" for large $x$, i.e. stretch the axis, you'd set $x=10^{x'}$ and thus $$y' = y = f(x) = f(10^{x'})$$
This is good, if $f'$ is very small.
If $f$ is very small in change, you could go for $y' = 10^y$ instead so
$$y' = 10^y = 10^{f(x)} = 10^{f(x')}$$
For very simple functions, say $f = {\rm id}$, you can interchange these equivalently with $\log$'s of the "other axis".
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 $(2cm, 3cm)$.
In short, you'll draw the same three points.
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...
Best Answer
On a log scale a relationship of the form $x \mapsto x^{\alpha}$ shows up as a straight line (of slope $\alpha$) when you plot $\log x^{\alpha}$ against $\log x$. This is useful with many data sets such as frequency responses in engineering, as you can easily estimate constants just by looking at their graph.
See Darrell Huff's 1954 classic "How to Lie with Statistics" for nice examples of misleading scales.