I’m unable to detach my mind from the linear scale imbedded in my head, how to make sense of a logarithmic increase?
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Just adding a thought to the nice explanation already given: if you are dealing with numbers on a big range, you prefer showing them on a log scale. This goes in both directions of “range”: far from 1 to the right and far from 1 to the left towards non-negative numbers. Between 1 and 1000 you got 1000 times bigger numbers, the same ratio as between 1 and 0.001. However if you plot on a linear scale, the 1000 is 1000 notches to the right in your plot, while the 0.001 is virtually just one notch left of 1. To “spread” them fairly, you plot them as fractions of each other, that’s what’s the log scale is doing.
Take some graphing calculator. Throw up two functions: y=x and y=2^x
Now, stretch the paper it’s drawn on so that the 2^x line is straight. Your original y=x line will be very bent. Your new ‘paper’ is a log scale.
These scales are useful, broadly, when comparing multiplication instead of addition. Going up one line in a linear scale means something is bigger by 1 (or some other fixed number), but on a logarithmic scale it means *multiplying*.
Say you want to compare the height of an ant, a human, and the burj khalifa. On a linear scale, the ant and the human are right next to eachother, while the BK is *really far away*.
This can be sort of misleading, though, as the ant is 500x smaller than the human, who is 450x smaller than the BK. An ant looking up at a human sees us towering above them, much like we see this huge skyscraper towering above us. The experience is similar; the size relationship between the ant and the human is like the size relationship between the human and the BK.
On a logarithmic scale, this is represented well. The ant and the human are about as far apart as the human and the BK, as opposed to the linear scale which would tell you that an ant and human are “about the same size”.
On a linear scale or graph, the measurements (ticks/ hashes) are usually measured in units. Comparing two plot points or values is generally how many more or less compared each other. (Addition or subtraction)
Point A = 5 and Point B = 8. From zero, to where you find 5, you get a sense of scale of how far 2×5 would be. Point B isn’t quite that far. A graph makes it intuitive roughly what value point B would have if it wasn’t labeled as 8.
With log scales, the comparisons are more like a proportion or factor comparison. (Multiply / divide)
Often a log scale is based on a factor of 10. Meaning, instead of -adding- 10 units for a certain distance, you’d multiply by 10 instead.
One way to help make sense of it is to think of each number in the scale as how many zeros it has.
The Richter Scale for measuring earthquakes is logarithmic, so an earthquake that measures 8.0 on the Richter Scale is ten times more powerful than one measuring 7.0.
10 = 1,000,000,000
9 = 100,000,000
8 = 10,000,000
7 = 1,000,000
6 = 100,000
5 = 10,000
4 – 1,000
3 – 100
2 = 10
1 = 1
In addition to the other replies here about putting a log scale on the y-axis, there are also times when it makes sense to put the log scale on the x-axis as well (known as a log-log scale). These can be useful in cases where, for example, you have a lot of data on the past, but lots of the interesting changes are taking part in the last (most recent) data.
Here are some interesting charts to compare, all showing the exact same data: world population from 10000BC to 2000AD.
[Linear scale:](https://upload.wikimedia.org/wikipedia/commons/b/b7/Population_curve.svg) looks essentially flat for thousands of years, some stuff starts happening around 1AD but it still looks fairly insignificant, and then there is a huge spike right at the end, but you can’t really tell what year the spike really started.
[Log scale:](https://upload.wikimedia.org/wikipedia/commons/f/f2/World_population_growth_%28lin-log_scale%29.png) you can see a lot more detail of the population fluctuations thousands of years ago, that all looked like “0” on the linear scale. There is still a massive spike right at the end of the 20th century, but you still can’t see down to what’s happening decade-to-decade.
[Log-log scale](https://upload.wikimedia.org/wikipedia/commons/8/84/World_population_growth_%28log-log_scale%29.png): now the thousands of years of data from long ago are compressed (but you can still see the fluctuations clearly), and the data is more zoomed-in as you get more recent- since more recent trends are probably what we’re more interested in.
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