# Proof of basic logarithm property

• Mar 17th 2013, 09:15 AM
GeneralFailure
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;)
• Mar 17th 2013, 10:16 AM
MINOANMAN
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*}