Log scales can introduce logarithms swimmingly! Let's start with a germane xkcd comic

Thanks to user Nat for charting the log-10 version in the comments! But I matched Nat's colors to mine.
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Imagine you wanted to plot the net worth of people in your area, in a graph that shows all the data. Presuppose that most net worth will be around the lower end, with some higher in between. And presuppose that the wealthiest person possesses $500K. You should be able to effortlessly see the distribution of net worth among your population.
Then suddenly Elon Musk decides to join your area. Now your graph is no longer readable, and it's warped! Why is it misshapen? Because the number line suddenly jumped to 147 billion!
As your graph is supposed to show all points, now it's squished such that Elon Musk is at the top, a single outlier in the whole distribution. Even the richest resident in your area is a measly 0.00034% on the scale — This is how laughable 500k is in that distribution!
Elon Musk's billions make it impossible to see any difference between the 500k guy, the 300k guy, the 30k guy, and so on. They're all down in the same line from Elon's perspective.
So now you take a logarithmic scale, meaning you take their net worth, throw it into a logarithmic function and then plot that value. You obviously have to adjust the y axis accordingly.
The 30k guy is now a 4.47.
The 200k guy is now a 5.3.
The 500k guy is now a 5.7.
And Elon? Well he's an 11.17.
These now fit on a single graph, and you can still make out differences between the "smaller" numbers.
Logarithmic scales are generally often used when exponential growth is concerned, like population growth, spreading of diseases. Also when you have to work with data that spans a huge range of values, like comparing the size of objects in the universe.
Machine learning and data science commonly use logarithmic scales and transformations.
Suppose you measure something that produces only positive numbers (this is a requirement to use a logarithmic scale). You get these measurements:
$\color{fuchsia}{1, 3, 5134573435345, 0.0000000053453}$
Ouch! That's quite a lot of variation there! If you plot this on a normal scale, the plot is going to look misshapen! One point is way off somewhere, and the rest sort of clustered around zero. By looking at this plot, nobody is able to tell that there's a huge difference between 1 and 0.0000000053453.
This is not a contrived example. I made up these numbers, but lots of natural phenomena are like this.
So what do we do? We can use a logarithmic scale. If I take the logarithm of the four numbers above, I get:
$\color{fuchsia}{0, 0.477, 12.7, —8.27}$
If you plot these numbers on a normal scale, things are much clearer! There's a negative number, 0, another near zero, and a positive number...all in the interval of $[-15, 15]$ that you can plot by hand!
Logarithmic scales are used for all sorts of phenomena that tend to produce numbers such as the ones above, such as earthquake intensity (Richter's scale), concentration of ions (pH-scale), anything that involves our senses (f-stops in photography, decibels, octaves), entropy, star brightness... the list goes on. Another less obvious example is the "five standard deviations" rule of thumb used by CERN, when CERN were looking for the Higgs boson.
Here are more real life applications of logarithms.