Alright, let’s talk about a better way of measuring how something has changed than just using percentages. When you first learn about percentages, they seem like a pretty nice way to measure how things change. Let’s say something increases in price from $10 to $15, you would say that it has increased in price by 50%. Or if something decreases in price from $30 to 20$, you would say that it has decreased in price by about 33%. So, what’s so bad about this way of quantifying things?
Well, consider a different example. Let’s say something costs $50 and then increases in price to $60. We can calculate how much it’s increased in price by calculating Now, let’s say something costs $60 and then decreases in price to $50. We can calculate how much it’s decreased in price by calculating . This should annoy you! Be annoyed. There’s an asymmetry here. When something increases by , in order for it to get back to its original value, it has to decrease by .
What if we could fix this? What if we could find a way to describe how much something has increased so that when you apply that same description to how something decreased you got a number that was related? What a nice world that would be!
Well…if we use logarithms, we can make this symmetric! Let’s go back to our most recent example. Suppose something increases in price from $50 to $60. Then we can describe this change using logarithms by saying that it increased by logarithm points. Now, if something decreases from $60 to $50, we can say that it changed in price by , or decreased by logarithm points. Now these two changes are described in a symmetric way!
Another nice feature of using logarithms to describe changing values is that it makes successive change easier to describe as well. Here’s an example. Let’s say something increases from $5 to $6 and then from $6 to $7. You say that it increased 20% and then 16.7%, but that doesn’t really give you an intuition about the total change. A better way to do this would be to say that it increased logarithm points. Then some cancellation happens, and we can see that the increased overall logarithm points.
I hope this showed you why logarithms can be useful when you want to quantify how much things have changed, and also gave a more intuitive way to think about percent changes that’s symmetric for increasing and decreasing values!
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