I'd like to understand the difference between linear, semi-log, and log-log plots.
This is one of those questions where I spent so much time trying to craft the question that I ended up understanding the answer to my question (I wrote this part after having written all of the below).
I'd like to know if the reasoning below is correct.
Consider the function
$$y=f(x)=10^x\tag{1}$$
Since the image of this function is $y>0$ we can take the base 10 log of each side of the equation
$$z(x)=\log{(y)}=x\tag{2}$$
which is a linear function of $x$.
If we plot this as is, we get a very simple plot
That is, if we plot $z$ vs $x$ we get a linear graph.
If we consider the values $z_1=1, z_2=2, z_3=3, ...$ on the vertical axis they are actually $z_1=\log_{10}{(y_1)}=1, z_2=\log_{10}{(y_2)}=2,...$, which means that these values are associated with values $y=10, y=100, y=1000$.
In addition, we have the negative values $z_{-1}=\log_{10}{(y_{-1})}=-1, z_{-2}=\log_{10}{(y_{-2})}=-2, z_{-3}=\log_{10}{(y_{-3})}=-3,...$ and these correspond to $y_{-1}=10^{-1}=0.1, y_{-2}=10^{-2}=0.01, y_{-3}=10^{-3}=0.001$, and so on.
$y$ never reaches 0, as expected, even though $z$ can become as negative as we would like.
We can keep expanding the graph downward as $y$ approaches zero, or upward as $y$ approaches infinity.
If we sub in the values of $y$ associated with each $z$ we get an axis that has uneven spacing of y values. That is the difference between $y$ and $y+1$ on the axis becomes progressively smaller as we go up the scale.
The reason is simply because $\log{(y+1)}-\log{y}$ is decreasing in $y$.
What about linear $y$ and logarithmic $x$?
We started with $y(x)=10^x$. $x$ takes on all values in this function.
Now, if $z(x)=\log{(x)}$, for $x>0$ then $x=10^z$ for all values of $z$ and so
$$y(z)=10^{10^z}\tag{3}$$
$$y(\log{x})=10^{10^{\log{x}}}\tag{4}$$
If we plot (3) we get a graph that blows up much faster than $10^x$ because now if we consider each value $z_1=1, z_2=2, z_3=3$ what we have is $z_1=\log{x_1}=1 \implies x_1=0$, $z_2=\log{x_2}=2 \implies x_2=100$, $z_3=\log{x_3}=3 \implies x_3=1000$.
If we label the points on the horizontal axis with their x-values, then we get the typical result we see when we see a graph with a logarithmic scale.
The value of $x$ never reaches $0$ and we have uneven spacing once again between $x$ and $x+1$ because the horizontal axis has values of $\log{x}$ and as we've seen $\log{(x+1)}-\log{(x)}$ is decreasing in $x$.
For this example, the easiest and final case is a log-log plot, where we have
$$y(x)=10^x$$
$$\log{y}=\log{(10^x)}=x=10^{\log{x}}$$
The shape of this plot is the same as that of the initial plot $y(x)=10^x$. There is the details that the values will also be the same on both axes if we are considering as variables $\log{y}$ and $\log{x}$.
The underlying values of $y$ and $x$ will each be "growing much faster than before" on this plot and the spacing will be uneven for these variables because what is plotted is actually their logs, which, once again have the characteristic that the difference between $\log{(z+1)}-\log{(z)}$ is decreasing in $z$.
At this point, I think I answered a lot of the questions I had when I had about logplots.
Yup, that all looks good. Glad you were able to answer your doubts. Sometimes, just writing out a question or explaining your doubts to a friend helps clear things up. In practice, the choice between plots depends on the scale of your data: if one variable ranges in value by one or two orders or magnitude, then the ordinary linear axis should be fine. But if the range is over several orders of magnitude, or you want to focus on how the plot changes over orders of magnitudes, then the logarithmic axis is better to effectively change the spacing between points.