Let $f$ be the function:
$$ f(t) = \begin{cases} \log{\left| t\right|} \quad 0<\left| t\right| \le 1\\ 0 \quad\quad\quad \left|t\right| > 1 \end{cases} $$
I have to prove that it admits the Fourier's transformation of type
$$ \hat{f}(\omega) = \int_{\mathbb{R}}{f(t)e^{-j\omega t}\,\mathrm{dt}} $$
And my notes says that it is sufficient if I prove that $f$ is absolutely integrable, but I can't find an easy way of proving that
$$ \int_{-1}^{0}{\left| \log{(-t)}\right| \mathrm{dt}} + \int_{0}^{1}{\left| \log{t}\right| \mathrm{dt}} \,<\, \pm\infty $$
I would appreciate any clue or method for solving this, thanks.