How to calculate Fourier Transform of logarithmic function?

Question

Given a random variable (RV) $S$ equal to the sum of two mutually independent (RVs) $X_1,X_2$,i.e.$S=X_1+X_2$ and piece-wise probability density functions (PDFs) of $f_{X_1},f_{X_2}$ are as follow: $$f_{X_i}(x) = \left\{ \begin{array}{l l} \displaystyle \frac{1}{d_i}\cdot \log\frac{\alpha_i}{x} & \quad \text{if $x \in (\alpha_i,0)$}\\ \displaystyle \frac{1}{d_i}\cdot \log\frac{\beta_i}{x} & \quad \text{if $x \in ( 0,\beta_i )$} \end{array} \right.$$ where $$d_i=\beta_i - \alpha_i, i=1,2$$ The PDF of $S$ is the convolution of $f_{X_1}$ and $f_{X_2}$, i.e. $$f_S(x)=f_{X_1}(x)\ast f_{X_2}(x)$$ However, how to get the Fourier Transform ( or characteristic function ) of above PDFs? Are there other orthogonal basis functions to transform the above PDFs into another space to help compute $f_S(x)$?