Minimum of $\sum\limits_{i=1}^n \ln(x+a_iy) -b\ln x + \sum\limits_{i=1}^n \frac{c_i}{x+a_iy} - \frac{d}{x}$ for $x, y>0$

Question

I have the optimization problem $$\begin{array}{ll} \min\limits_{x, ~y} & \displaystyle\sum_{i=1}^n \log(x+a_iy) -b\log(x) + \sum_{i=1}^n \frac{c_i}{x+a_iy} - \frac{d}{x}\end{array}$$ where $x\geq \varepsilon$ and $y \geq0$ are the decision variables, for a constant $\varepsilon>0$ $a_i,b,c_i,d \geq 0$ are constants.
I am wondering how to reformulate this problem so I can solve it.

If I only have the two rightmost terms, I can reformulate the problem using the usual Geometric Programming techniques, but I do not know how to handle the logarithms within this framework. Any help will be appreciated.