Note: Assuming base of logarithm as $e$.

Given 

Assertion$:$ for $x<0, \frac{d^2}{d{x}^2}\left(\log \left|x\right|\right)=\frac{1}{|x{|}^2}$

Reason$:$ for $x<0, \left|x\right|=-x$

So, let $f\left(x\right)=\log \left(\right|x\left|\right)$

For $x<0$

$y=f\left(x\right)=\log (-x)$

Now, $\frac{dy}{dx}=\frac{1}{(-x)}\left(-1\right)=\frac{1}{x}$

$\frac{d^{2}y}{dx}=-\frac{1}{x^2}$

So, Assertion is wrong but reason is true.