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Math Help - Proof of basic logarithm property

  1. #1
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    Proof of basic logarithm property

    Hi everyone I'm new here and as almost every newbie have come here with a problem:

    I have defined lnx as \lim_{ n \to \infty }n( \sqrt[n]{x} - 1 ))
    Then i proved that the sequence cinverges and some basic properties: it is inverse function of e^x and sum and subtraction. However I'm having trouble prooving formally that
    \ln{x^a} = a\ln{x} for any real a( for natural numbers the proof is trivial of course). I suppose there would some kind of transformations using limits but i can't do it. Any help will be appreciated.

    Thanks
    Last edited by GeneralFailure; March 17th 2013 at 10:19 AM.
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  2. #2
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    Re: Proof of basic logarithm property

    check here for more information specially the part for Analytic properties....

    Logarithm - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Proof of basic logarithm property

    I took a look. However there is no formal proof of what i want:X Thanks anyway
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  4. #4
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    Re: Proof of basic logarithm property

    If you know that \left(x^y\right)^z=x^{yz} and e^x is the inverse of \ln x, and if you can use Taylor series, then

    \begin{align*}\ln\left(x^a\right)&= \lim_{n\to\infty}n(x^{a/n}-1)\\ &=\lim_{n\to\infty}n((e^{\ln x})^{a/n}-1)\\ &=\lim_{n\to\infty}n(e^{(a\ln x)/n}-1)\\ &=\lim_{n\to\infty}n(1+(a\ln x)/n+o(1/n)-1)\\ &=\lim_{n\to\infty}a\ln x+o(1)\\ &=a\ln x\end{align*}
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