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Here is a comparison of CO2 infrared absorption as postulated by arrhenius in 1896, as measured by arrenius in 1901 and calculated by modtran using modern spectra:
I verified the claim by Weart that arrhenius reduced his value of 5 degres for CO2 doubling in 1901.
Like so many others you are confusing cause and effect the response of CO2 to temperature is approximately linear 10 ppm/K the response of temperature to CO2 is climate sensitivity which is the logarithmic arrhenius law, which parameters are under heavy debate.
You'll find out, like arrhenius did, that CO2 is a greenhouse gas.