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Thread: e

  1. #1
    Forum Sophomore
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    Hello community! I am mastermind. Hope I will find some pleasure here and can discuss many things.
    Here my question now:

    Why exactly is e called the "natural" logarithm?

    And why is e^z = 1 + z/1 + (z^2)/2 + (z^3)/3 + ... ?

    Thank you for help!


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  3. #2  
    Forum Professor river_rat's Avatar
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    Hi mastermind

    1. e is not the natural logarithm, it is the natural base for exponents in calculus though. e<sup>x</sup> = 1 + x + x<sup>2</sup>/2! + x<sup>3</sup>/3! + .... and not what you posted btw. You forgot the factorials in the denominator
    2. Log(1+x) = x - x<sup>2</sup>/2 + x<sup>3</sup>/3 + .... is the expansion for the natural logarithm that you asked for.
    3. Both of these are examples of taylor expansions, do you know what those are or should i go through the steps?
    4. The "natural" statement comes from the fact that if f(x) = e<sup>x</sup> then f'(x) = e<sup>x</sup> and this is the only base for which this statement holds. So it is natural to use this base (as it is natural to use radians for measuring angles) as you dont have messy constants popping out of your equations.


    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  4. #3  
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    My own belief has always been that it is called 'natural' because it is used throughout formula dealing with growth and decay, that is 'e' can be used to predict the growth of humans, tree's, the natural cooling of water.. etc.
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  5. #4
    Forum Professor leohopkins's Avatar
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    I thought that e's were drugs ?
    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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  6. #5 Re: e 
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    Quote Originally Posted by leohopkins
    I thought that e's were drugs ?
    From the quality of your post's I'd say you have a considerable knowledge of the subject! 8)
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  7. #6 Hey...... 
    Forum Professor leohopkins's Avatar
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    Hey, it wasnt ME that said that quarks could come in "flavours"!.
    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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