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E (with Ln)

  1. Jan 27, 2007 #1

    JPC

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    hey

    i know what the log function is

    log a (x) : can be translated to :
    a ^ y = x
    with y to find

    but with the Ln its
    ln(x) = e ^ y = x
    with y to find

    and e = about 2.7....

    But why this number ?
    whats special about 2.7.... ?
    whats the exact value of e ?
     
  2. jcsd
  3. Jan 27, 2007 #2

    arildno

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    The exact value of e is e.

    Strategies to convert that value into its equivalent decimal representation abound, the most common centering on the defining identity of e:
    [tex]e=\sum_{n=0}^{\infty}\frac{1}{n!}, 0!=1, n!=n*((n-1)!), n\geq{1}[/tex]
     
  4. Jan 27, 2007 #3
    Goodmoring ALL,
    the number e as "arildno" e=e and e can be calculated by an infinite series and it is a Real number.
    But you asked why e?
    we must know that e is the contraction of eulur, and was a mathematics scientist and he had make a great researches aboiut the number e.
    also, what special about e?
    physicains had found many phenomenans the changes by a function f(x) = log a(x) [log to the base a]
    and the found that the aproximate value of a =2.7.....
     
  5. Jan 27, 2007 #4
    Ever heard that [itex]ln[/itex] is sometimes called a natural logarithm.
    Why would they call that natural and what a heck does [itex]e[/itex] have to do with nature you may ask.
    Instead of answering this in a long and bit a philosophical way ,I will ask you quite a similar question :Why [itex]\pi[/itex] ,and why is important about that constant.I will not answer neither of two questions leaving you to ponder over them alone.
    If you ask why is [itex]e=2.71...[/itex]and [itex]\pi=3.14...[/itex] than the answer is simple:That's becouse people like to use decimal number system (and I guess I know why ,when I take a look at my hands).
    The last question "what's the exact value of e"?.The most difficult one.
    Well I think I will not dare answering that one...
     
    Last edited: Jan 27, 2007
  6. Jan 27, 2007 #5

    arildno

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    I've dared:
    The exact value of e is e. :smile:
     
  7. Jan 27, 2007 #6

    mathwonk

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    the exact value of e is: the smallest real number larger than all finite sums of the series:

    1 + 1 + 1/2! + 1/3! + 1/4! + 1/5!+.............

    it is also exactly the unique x value such that the area under the graPH OF Y = 1/X, FROM X=1 TO x=e IS 1.

    it is also probably the unique smallest real number larger than all the powers of form (1+ 1/n)^n, for all positive integers n.

    it is approximately equal to 2.718281828459......

    e is the unique positive real number a such that the derivative of the function a^x is a^x.

    e is the value at x = 1, of the unique solution of the equation

    f' = f, f(0) = 1.

    e is the unique positive real number a such that the function f(x) = a^x has derivative at zero equal to 1, i.e. such that f'(0) = 1.
     
    Last edited: Jan 27, 2007
  8. Jan 27, 2007 #7
    Yes ,yes .Of course,that is an alphabetical number system !
    How could I forget that one...:smile:
     
  9. Jan 27, 2007 #8

    arildno

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    Two point seven, two times Ibsen, 459045...:smile:
     
  10. Jan 27, 2007 #9

    D H

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    Why e?

    "Real and Complex Analysis" by Walter Rudin starts with a prologue on the exponential function. The first sentence is "The exponential function is the most important function in mathematics."

    Among other very useful features,
    • The exponential function is the only function (to within a multiplicative constant) whose derivative is equal to itself.
    • The exponential function is related to the trigonometric functions via the Euler formula,[itex]\exp(ix) = \cos x + i\sin x[/itex].
    • The derivative of the inverse of the exponential function, [itex]\log x[/itex], is simply [itex]1/x[/itex].
     
  11. Jan 27, 2007 #10
    The natural logarithm ln(x) is the simplest of the logarithms. It's defined as the area (or integral, if you will) under the graph y=1/t when t goes from 1 to x. Specifically, when the area is 1, ln(x)=e (just like mathwonk said). Also, the inverse function of ln(x) is e^x, which you may know is the only function which is its own derivative (well, it can be multiplied by a constant too), which makes it even more special.
     
  12. Jan 27, 2007 #11
  13. Jan 27, 2007 #12

    arildno

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    Euler himself used "e" as signifying the exponential function.
    Only fifth-rate mathematicians assign their own name to their objects of study, and Euler was top-of-the-notch.

    Noone knows why he called the number "e"; it is perhaps most likely it originally was short-hand for "that number which serves most "naturally" as the base in Exponential functions".
     
    Last edited: Jan 27, 2007
  14. Jan 27, 2007 #13

    HallsofIvy

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    Any exponential function ax has the property that its derivative (rate of change) is just a constant times itself: Cax. You can show that for a= 2, that number is less than 1 but that for a= 3 it is larger than 1. There exist a number between 2 and 3 such that the constant is exactly 1. That is the number we call e. The function ex has the property that its derivative is simply ex itself.
     
  15. Jan 28, 2007 #14

    JPC

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    What does the : '!' mean ?
     
  16. Jan 28, 2007 #15

    arildno

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    See post 2, for definition of n!
     
  17. Jan 28, 2007 #16

    cristo

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    "!" means factorial: 5!=5*4*3*2*1. So, in general n!=n(n-1)(n-2)...1.
     
  18. Jan 28, 2007 #17

    JPC

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    thanks

    i tested with a program :
    with ent.text : sum of all from 0 to ent
    res.tex : the result

    Dim c As String
    Dim d As String
    Dim f As String
    Dim a As String

    f = "1"
    a = "1"

    c = 1

    Do Until c * 1 = ent.Text * 1
    a = a * 1 * c * 1
    d = 1 / a * 1


    f = f + d * 1

    c = c + 1 * 1

    Loop

    res.Text = f



    and result is good
     
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