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To start with, I need to write a list of assumptions that are at play, and numerous disclaimers are needed. Firstly, the IPCC scientists don't say this follows a $ln$ function at all. They say it follows whatever their computer models says it follows. This is only a first order solution. Assumptions begin here:

$$F_{CO2} = \int_0^{\infty} \beta \Phi (1 - e^{\tau(\lambda)}) d\lambda$$

$$(1 - e^{\tau(\lambda)}) \approx \begin{cases} 0, & \mbox{if } \tau(\lambda) < 2.3 \\ 1, & \mbox{if } \tau(\lambda) \ge 2.3 \end{cases}$$

To start with, I need to write a list of assumptions that are at play, and numerous disclaimers are needed. Firstly, the IPCC scientists don't say this follows a $\ln$ function at all. They say it follows whatever their computer models says it follows. This is only a first order solution. Assumptions begin here:

$$F_{CO2} = \int_0^{\infty} \beta \Phi (1 - e^{-\tau(\lambda)}) d\lambda$$

$$(1 - e^{-\tau(\lambda)}) \approx \begin{cases} 0, & \mbox{if } \tau(\lambda) < 2.3 \\ 1, & \mbox{if } \tau(\lambda) \ge 2.3 \end{cases}$$

The reference for this work was posted by Michael Luciuk in the comments. http://www.princeton.edu/~lam/documents/LamAug07bs.pdf

And people have, of course, pointed out that the real behavior doesn't exactly follow this model. Here is some real data. I found it to look surprisingly logarithmic. Source

Now here is the beef, we can find $\Delta \lambda_1$ from the prior Taylor series expansion. Remember, the edge of the absorption peak, the point that separates the optically thick and thin regions, comes from that magic 2.3 value. The driving force behind this is, of course, the change in concentration $C$. For clarity I'll note that $\Delta \lambda_1 = \lambda_1-\lambda_1'$.

Now here is the beef, we can find $\Delta \lambda_1$ from the prior Taylor series expansion. Remember, the edge of the absorption peak, the point that separates the optically thick and thin regions, comes from that magic 2.3 value. The driving force behind this is, of course, the change in concentration $C$. For clarity I'll note that $\Delta \lambda_1 = \lambda_1-\lambda_1'$. You'll probably be confused by the negative, it's just math trickery in order to account for the fact that we're on the left (negative) side of the peak, and thus get the right $C$ on top.