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