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Exponent Law proof Cant find anywhere?

  1. Oct 26, 2011 #1
    I have been studying the exponent laws in depth and I cannot find a formal proof on this law:

    exp(a+b) = exp(a)exp(b)

    Is it formal enough to say the following? (I think not since I give an example with numbers in it)

    We define:
    a² = a*a.
    For example:
    a^8 a^7 = a*a*a*a*a*a*a*a a*a*a*a*a*a*a = a^15 = a^8+7

    More generally:
    a^(c+b) = a^c * a^b = a^(b+c)

    Regards as always
    Tam
     
    Last edited: Oct 26, 2011
  2. jcsd
  3. Oct 26, 2011 #2

    lurflurf

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    Homework Helper

    Proofs of this type depend upon your definition of exp.

    I like to take the addition formula as the definition, so the proof is obvious from the definition.

    Others define exp with a limit, a series, an inverse of an integral, or a differential equation and prove it by manipulating whatever their definition is.

    Your proof can be extended to prove it for all rationals. The extension to reals is done by taking limits
     
  4. Oct 26, 2011 #3
    Thanks for your answer.

    I'd like to take the proof from this definition: exp(x) = lim (n→∞) (1 + x/n)ⁿ (Limit)

    Would you be able to point my in the right direction please?

    Regards
    Tam
     
  5. Oct 26, 2011 #4

    HallsofIvy

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    The best formal proof would be by induction.
     
  6. Oct 27, 2011 #5

    lurflurf

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    Try to show that
    exp(a)exp(b)/exp(a+b)=1
    starting with
    exp(a)exp(b)/exp(a+b)=
    lim (n→∞) (1 + a/n)ⁿ lim (n→∞) (1 + b/n)ⁿ /lim (n→∞) (1 + (a+b)/n)ⁿ =
    lim (n→∞) [(1 + a/n)(1 + b/n)/(1 + (a+b)/n)]n
    and so on
     
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