Trying to prove logarithms preserve limits without any notion of continuity

I am trying to prove that if ${s_n}$ is a convergent sequence, $\lim_{ n \to \infty} s_n = s$ iff $\lim_{ n \to \infty} \log(s_n) = \log(s)$. I don't have any notion of continuity yet (although I know that continuous functions are precisely the ones that preserve limits), so i am trying to do this without that fact. This is part of a larger problem where i have a sequence $s_n$ which i am trying to show converges to 1, but so far i managed to show $log(s_n)$ converges to $0$.

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The fact that the logarithm function is monotone should come in very handy. So should the fact that $\lim_{n\to\infty} x^{1/n}=1$ whenever $x>0$, I think. –  Harald Hanche-Olsen Nov 14 '12 at 8:57
Well, what definition of $\log$ are you using? –  Rahul Nov 14 '12 at 10:16
Mike: Maybe by looking at how you got to log(s_n) --> 0 you can "work backwards" and see how to actually get s_n --> 1 on reversing the steps. –  coffeemath Nov 14 '12 at 18:17