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I'm looking to extrapolate rising temperatures based on the increasing CO2 levels. From this article: CO2 in 2050, with the following formula used to extrapolate the CO2 levels:

$CO_2\ level = 280 \cdot \left( 1+\exp(0.0222\cdot \left(year-2052\right)) \right)$. This formula gives the projected CO2 level in parts per million (ppm) for any year from 1958 on, assuming the growth rate is constant at 2.22%.

This seems fair since CO2 increases by 2.2% per year and it's effects are exponential.

I want to roughly use the same formula to extrapolate future temperatures based on this same CO2 increase as the accompanying document suggests: Temps using CO2. The issue is the constants used... because if the same formula is used, then at year 2052, the temperature would be doubled which is obviously not realistic.

So my question is: what coefficients should be used in this: $T=\left(1+\exp(ct)\right)$ formula, where $c$ is a constant ($0.0222$ I would think) and $t$ being a some measure of time.

Or is there a better formula to use which correlates exponentially increasing CO2 with increasing temperatures?

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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.

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