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I'm supposed to prove that :

For $g(x) = a_0+a_1 \cdot x+\cdots+a_n \cdot x^n$ a polynomial of degree $ n$ where $ n \ge 0$ and $a_n \ne 0$.

Prove that $\log|g(x)|$ is $O(\log(x))$.

I've been able to do it for specific polynomials, but I can't seem to prove this with a generic polynomial.

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We can assume that $a_n=1$. Let $R$ such that if $|x|>R$ then $\left|\frac{a_0+\dots+a_{n-1}x^{n-1}}{x^n}\right|\leq 1$. Then $\log|g(x)|=\log\left(|x|^n\left(\left|\frac{a_0+\dots+a_{n-1}x^{n-1}}{x^n}\right|\right)+1\right)\leq \log(|x|^n+1)\leq \log 2+n\log |x|$ if $|x|>\max\{R,1\}$. This gives the wanted result, as if we assume $|x|\geq \max\{R,2\}$, we get $\log|g(x)|\leq (n+1)\log|x|$.

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    @martycohen Yes, for example when $g(x)=x^n$ we have to take a constant $\geq n$.2012-10-07