I'm trying to interpret the slope and $y$-intercept of a rather specific equation. The equation I'm dealing with is $$R = C\left(\frac{L}{10^{44}\text{ erg s}^{-1}}\right)^\alpha$$ taking the base-10 logarithm of both sides, I get $$\log_{10}(R) = \left[\log_{10}(C)-\alpha\log_{10}(10^{44})\right]+\alpha\log_{10}(L)=K+\alpha\log_{10}(L)$$ also known as the Radius-Luminosity relation for supermassive black holes in astrophysics, and is linear in log space. The best-fit values for the $y$-intercept $K$ and slope $\alpha$, with their quoted uncertainties, are given by $$K=-21.3^{+2.9}_{-2.8}\qquad\text{and}\qquad\alpha=0.519^{+0.063}_{-0.066}$$. Therefore the coefficient $C$ in the first equation is given by $$C = 10^{K+\alpha\log_{10}{10^{44}}}$$
Plugging in the values for $K$ and $\alpha$ yield a value of $C=34.36$. I would like to determine the maximum and minimum values for the coefficient $C$ given the maximum and minimum possible values for the $y$-intercept and slope $\alpha$. I do not think it is as simple as plugging in $K\pm\delta k$ and $\alpha\pm\delta\alpha$ (the values which would maximize/minimize $C$), since this would produce unrealistic upper and lower limits to $C$ ($10^{7.208}$ and $10^{-4.168}$ respectively).
Instead I think the value of the slope determines the uncertainty in the $y$-intercept. That is to say a higher value of $\alpha$ corresponds to a lower value of $K$. This makes sense; if we increase the slope the $y$-intercept gets shifted down. Then if the maximum value of $\alpha$ corresponds to the lowest value for $K$, I get $C=32.21$ for a minimum value of $C$, and vice-versa, taking lowest possible value for $\alpha$ and the highest possible value for $K$, I get $C=34.04$ for a maximum(?) value for $C$.
So my maximum value for $C$ is still less than my optimal value of $C=34.36$. Obviously I did something wrong but I can't figure out what.
Is there a way to determine the maximal and minimal values of $C$ from $\alpha$ and $K$? And if so, how?
You should write $LogR=LogC+\alpha(LogL-44)$ and do the linear regression on that