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Many things in the real world are approximately fractal or logarithmic. For example, if you examine a shore line it will be a wriggly line. Examine it at more detail and you will see a similar pattern but at a smaller scale. Even more detail and you still have the same (or similar) pattern at yet more detail. Computer-aided graphics use this property to generate landscapes: storing a small amount of "data" and replicating it at different scales is far easier than storing masses of data.
The logarithmic function also has this scale-invariant property. If you are interested, read the attached link about Benford's Law. The article does not require much mathematical knowledge - only curiosity.
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