How do I find the logarithm and exponential of a string of powers of x (including non-integer powers) in terms of x?

Question

I am trying to find a way to separate the negative and positive powers of x in the solution to the exponential and logarithm of real powers of x, where $M_L,M_E\subset \Bbb{R}$: $$y=\ln\left(\sum_{k \in M_L} q_k(x)^k\right),z=\exp\left(\sum_{k \in M_E} q_k(x)^k\right)$$ I currently know that: $$y=\ln\left(\sum_{k \in M_L} q_k(x)^k\right)=\sum_{n=0}^\infty \frac{(-1)^{n+1}}{n}\left(\sum_{k \in M_L} q_k(x)^k-1\right)^n=\sum_{k\in N_L} p_k (x)^k$$ $$z=\exp\left(\sum_{k \in M} r_k(x_k)^k\right)=\sum_{n=0}^\infty \frac{1}{n!}\left(\sum_{k \in M} r_k(x_k)^k-1\right)^n=\sum_{k\in N} s_k (x_k)^k$$ How can I separate y and z into positive and negative segments without having to approximate each series each time I need to calculate a different sequence of powers? $$z=z_+z_0+z_-,y=y_++y_0=y_-,$$ $$y_+=\sum_{k\in N_L\land k>0} p_k,y_-=\sum_{k\in N_L\land k<0} p_k$$ $$z_+=\sum_{k\in N_E\land k>0} s_k,z_-=\sum_{k\in N_E\land k<0} s_k,z_0=s_0$$ (Note: The purpose that I am using this for means I can find $y_0$ and $z_0$ by assuming x=1 and evaluating the functions. I only need to separate positive and negative powers of x). Any help would be appreciated!