Let $a,b,c_i,d_i\in\mathbb{R}$ be scalar coefficients, and let $\mathbf{x}=[x_{1},x_{2},\dots,x_{N}]$. Assume that I have two inequalities:
$$ \begin{align} &\sum_{i=1}^{N}c_{i}x_{i} &&\leq a \tag{1} \\ &\sum_{i=1}^{N}d_{i}\log x_{i} &&\leq b \tag{2} \\ \end{align} $$
My question: How would I find the intersection point between these two inequalities (if it exists)?
Some thoughts: If (2) would be a linear function like (1), then I could find the intersection point by solving a linear system of equations
$$\mathbf{A}\mathbf{x}=\mathbf{b} \tag{3}$$
where $\mathbf{A}=\left[\begin{matrix}c_1 && \dots && c_{N} \\ d_1 && \dots && d_{N}\end{matrix}\right]$ and $\mathbf{b}=\left[\begin{matrix}a \\ b \end{matrix}\right]$.
Likewise, if (1) were a sum of logarithmic terms such as (2), I could simply define $y_{i}:=\log x_{i}$, solve the resulting linear system of equations for $\mathbf{y}$, then get my solution as $\mathbf{x}=\exp\mathbf{y}$. It is only if the two types of equations are mixed that I run into problems. Can you recommend a strategy to resolve this?