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Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
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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