The proof is purely in the numbers.
If all the CO2 form the beginning of our atmosphere(~1ppm) to the level of 271.8 ppm, caused 30 W/m2 of energy imbalance
then the equation used by everyone, including the IPCC shows that the first 20 ppm pushed more than half of the total imbalance.
Since you will not believe me, let's run the numbers.
I will use the ACS's formula
Climate Sensitivity - American Chemical Society
in the case of all the CO2 5.35 X ln(271.8)=29.987 W/m2.
So the change from 1 ppm to 20 ppm, is 5.35 X ln(20)=16.027W/m2
and the rest of the CO2 above 20 ppm, is 5.35 X ln(271.8/20)=13.95W/m2.
So the graph I found from the Washington state government (and Cited),
is supported by numbers and formulas from the American Chemical Society.
So go ahead, and show me how CO2's forcing is not a natural log curve?