I am trying to convexify $\log(|x|)$. I think its concave. So I am trying to get an affine upper bound through linearization. But the problem is there are two concave functions because of the absolute value and affine upper bound for one may not hold true for another.
Any way I can construct an affine upper bound for this kind of concave functions
I am in slight trouble here and any help would be greatly appreciated
Clearly, your function can't be bounded by an affine function on the whole $\Bbb R$, because $\lim_{|x|\to\infty}{\ln|x|}=+\infty$ and because $\lim_{x\to 0}\ln |x|=-\infty$.
However, you can build a piecewise affine upper bound. For example, $\ln|x|<|x|$ for all $x\in\Bbb R$.