Question:
What is the intuition behind/mathematical purpose of
using logarithms and natural logs? I know the definition and the
algebraic rules of manipulting them, I just never got the why of using them.

Replies:
Logs help you answer the question "four to what power equals seven". What if
you were doing an experiment and you got the following data:

x y
1 1
2 4
3 9
4 16

You'd say "Oh, I get it. I bet if I set x to 5 I'd get 25." But you'll
probably never do a real experiment and get data with a pattern so easy to
recognize. You might get something like the following:

x y
1 1.00
2 4.92
3 12.5
4 24.3

At this point you could say "Seems like there's something simple going
on, but I don't quite get it.", or you could take ln(y)/ln(x) and get it
immediately.

Logs are good for making certain kinds of things easier to understand, and
for making certain kinds of problems easier to solve. They're just another
tool.

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