Formula for rising temperatures
I suggest you consider the concept of radiative forcing. It is the net change in energy balance at the top of the atmosphere due to imposed perturbation. The change in radiative forcing is a function of multiples of factors: water vapor, solar radiation, anthropogenic and natural emissions of CO$_{2}$, etc. So, attempting to quantify future temperature to a single forcing may not be a good way of relating the two vars. You may consider running climate models that consider the above factors. 
However, you can play around with single var relation you suggested. I can proved an alternative derivation that considers radiative forcing. 
The relation between temperature perturbation and radiative forcing is 
$$\Delta T = \lambda RF$$
where $\lambda$ is the climate sensitivity parameter and RF is a radiative forcing due to a particular perturbation. 

The above equation can be re-written,
$$T - T_{o} = \lambda RF $$
where $T_{o}$ is a particular reference temperature. You can take base period temperature (e.g., 1980 to 2015). 

Assuming temperature increases steadily in time, $T=T_{o}e^{ct}$, and inserting into the above equation,

$$T=\lambda RF (\frac{1}{1-e^{-ct}})$$
 Now you have $\lambda $ and  RF as coefficients.