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 $\displaystyle \lim_{ n \to \infty }n( \sqrt[n]{x} - 1 ))$

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

$\displaystyle \ln{x^a} = a\ln{x}$ 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;)

Re: Proof of basic logarithm property

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

Logarithm - Wikipedia, the free encyclopedia

Re: Proof of basic logarithm property

I took a look. However there is no formal proof of what i want:X Thanks anyway:)

Re: Proof of basic logarithm property

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

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