How much warmer will Earth get if carbon dioxide doubles? This number, the climate sensitivity, combines the forcing from carbon dioxide with all the feedbacks. Because the forcing grows with the logarithm of the concentration, the math uses logarithms.
Key idea
The forcing from carbon dioxide grows with the logarithm of its concentration: $$\Delta F = 5.35\,\ln\frac{C}{C_0}\ \text{W/m}^2$$ so each doubling adds the same push, about 3.7 W/m² by this formula (the IPCC's latest assessment puts it near 3.9). The equilibrium climate sensitivity, the eventual warming per doubling, is likely 2.5 to 4 °C, with a best estimate of 3 °C (IPCC 2021). So the eventual warming is about $S \times \log_2(C/C_0)$.
Why it works
Why a logarithm? Carbon dioxide absorbs infrared strongly near the center of its absorption band, so the center is already "full": adding more there changes little. Extra carbon dioxide mainly widens the band at its edges, where absorption is weaker, and that widening grows more slowly with each addition. The result is that each doubling, 280 to 560 ppm or 560 to 1120 ppm, adds about the same forcing. The warming arrives slowly: the deep ocean takes centuries to warm, so the warming at any moment lags behind the eventual (equilibrium) warming.
Worked example: One doubling
Carbon dioxide goes from 280 to 560 ppm. With a sensitivity of 3 °C per doubling, what is the eventual warming?
- $560/280 = 2$: exactly one doubling, $\log_2 2 = 1$.
- Warming: $3 \times 1 = 3$ °C.
Answer: 3 °C
Worked example: Where we are now
Carbon dioxide was about 425 ppm in 2024, compared with 280 ppm before industry. How many doublings is that, and what eventual warming does it imply at 3 °C per doubling? ($\log_2 1.52 \approx 0.60$)
- $425/280 \approx 1.52$, and $\log_2 1.52 \approx 0.60$ doublings.
- Eventual warming from CO₂ alone: $3 \times 0.60 \approx 1.8$ °C.
- The measured warming is so far about 1.1 to 1.3 °C, smaller because the ocean is still catching up and aerosols mask part of it.
Answer: About 0.6 doublings; about 1.8 °C
Common mistake
Expecting each added ppm to warm by the same amount. The push grows with the logarithm of the concentration, so going from 280 to 560 ppm warms about as much as going from 560 to 1120. Check: equal ratios give equal forcing: $5.35\ln 2 \approx 3.7$ W/m² for any doubling.