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next question: what is a natural log and what is the mathematical relevance of it.

Mathematics
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natural log is ln like ln(e)
Well it's a function like sin, cos, e
It has some interesting applications

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Other answers:

This is a long one, you first should know about e. So Euler had this question, is there a function that if you derive it, you'll get the same function ? So apparently yes, and he found the function to be e^x, where e is some weird irrational number :) e = 2,71828183 ... And the natural log is just the inverse of this e^x. So if you have y = e^x then x = ln(y). Of course there is so much more about this, I just cant explain it all in here :)
Another very interesting application comes from a limiting process... is it better to have your interest compounded annually, or bi-annually, or quarterly, or monthly... or weekly, or daily... or hourly... or every second, etc... TURNS OUT If you take this out to infinitely small time intervals you arrive at the number e. More here: http://tutorial.math.lamar.edu/Classes/Alg/ExpLogApplications.aspx
Continuous compounding.

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