Why are logarithmic graphs useful?
In: Mathematics
Logarithmic graphs are generally useful if you are dealing with data that varies across several orders of magnitude. Say, for instance you have some variable with values ranging from a few hundred to a few hundred thousand. If you try to plot the data on a linear plot, all your values at lower orders of magnitude get clustered together near the graph’s origin. Plotting this data on a logarithmic graph helps spread out those lower values and makes it easier to see trends that might be present.
They are very good to see growth. Especially in banking if you have something like a stock that is 10 times as valuable now than five years ago a 1% change now is larger than a 9% change then.
With logarithmic scales a 10% drop will be the same size on every magnitude. Try it out with bitcoin over it’s lifetime and you will be able to spot every fast value drop/increase even the ones at 1$
Log base 2 can be very useful for things that grow by dividing. A great example of this is cell growth.
Edit: The use in my example of that ac straight line on this graph would be that cells are both diving and dieing at constant (can be the same or different from each other) rates. A change in the slope of the line indicates a change in one of those rates which is useful to know. This is also hard to observe directly on a standard base 10 or log base 10 graph
In addition to the other reasons given, plotting functions with exponential behavior on log-log or semilog graphs can linearize them, making it easy to do simple graphical analysis on them.
When measuring something like how fast a website responds. I might want to see how fast it is for 50% of people and then the slowest 25% and then the slowest 1%.
The 50% might be 120ms, but the slowest 25% is 800ms, and the slowest 1% is 2000ms.
If I graph those on a normal graph variations in the 50% line are lost because of the scale needed to show the 1%. if I graph at a log scale I can see the detail for all 3 of those lines.
It’s the best representation of a wide magnitude of numbers. You can represent 1 to 1000 on the same graph via log
Graphs are useful because they show the relationship between two things. For example, the relationship between engine torque vs. engine rpm forms the familiar “bump” that can give us a lot of intuition about the characteristics of that engine.
Other relationships make for terrible graphs when using the same 1:1 ratio. For example the exponential growth in such things a cell populations or bank accounts, or sound loudness are difficult to show. They will look like a flat line before “suddenly” rising to the top. These graphs don’t give us any intuition, thew will all look the same. But if we adjust the ratio between axes to be logarithmic, suddenly the graphs will show the delicate nuances that let us form useful intuition about the characteristics of the values we are graphing.
Some subject matters just make more sense on a logarithmic scale.
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