Often when we discuss on the logarithms in high school we also talk about a scale called logarithmic.
In the he logarithmic scale: the distance from $1$ to $2$ is the same as the distance from $2$ to $4$, or from $4$ to $8$ as the image below.
There are many applications of the logarithmic scale: Weber-Fecner's law, sound intensity perceived by our hear hearings, the brightness of a star, etc.. What is the optimal solution or explanation for understanding how to make students build a logarithmic scale and its utility?
If I remember correctly, this video does a really good job at this sort of thing: https://www.youtube.com/watch?v=CfW845LNObM .
What you have shown above is just a different way of looking at what exactly a function does. For example, let's say we had
$${f(x) = x + 1}$$
This just "moves the number line left 1 unit" - but it does not affect the distance between two numbers. The numbers $0$ and $1$ have a distance of $1$ between them, and ${f(0)}$ and ${f(1)}$ has a distance of ${f(1)-f(0)=2-1=1}$ between them also - the distances stay the same!
Another example is
$${f(x) = 2x}$$
In this case, the distances between numbers are affected. Take the same example of $0$ and $1$ - ${f(0) = 0, f(1) = 2}$, and so ${f(0)}$ and ${f(1)}$ have a distance of $2$ between them - it's double the distance! And you can show that ${\left|f(y) - f(x)\right|=2\left|y-x\right|}$. The distance between numbers always get's doubled under this "transformation". So this function $f$ is a transformation that does stretch the number line.
So the Logarithmic scale is just showing you how the distances between two numbers get changed under ${\log(x)}$ as a transformation. And because ${\log(x)}$ grows so slowly - we expect distances between numbers under the transformation to get closer and closer together - just as you see in the picture.
We have some special names for how distances get changed too. Lipschitz mappings are mappings such that
$${|f(y)-f(x)|\leq k|x-y|}$$
For some constant $k$, and Contraction mappings are Lipschitz mappings with ${0\leq k<1}$. In fact one of the (in my opinion) coolest theorems ever is the contraction mapping theorem - but this is getting quite complicated :)