Approximation in Peskin's Chapter 6 In Peskin QFT's Chapter 6.4, IR divergence in 1st correction of interaction between electron and photon becomes
\begin{align}
F_1(q^2)\gamma^{\mu}+F_2(q^2)(p^{\mu}+p'^{\mu})
\end{align}
And
\begin{align}
F_1(q^2)\simeq \frac{\alpha}{4\pi}\int _0^1 d\xi \left[\frac{-2m^2+q^2}{m^2-q^2\xi(1-\xi)}\log\left( \frac{m^2-q^2\xi(1-\xi)}{\mu^2}\right)+2\log\left(\frac{m^2}{\mu^2}\right)\right]
\end{align}
in IR limit $\mu\to 0$ (This is equation just above (6.61)). And He said in the limit $\mu \to 0$, we can ignore the details of the numerators inside the log. And this becomes
\begin{align}
F_1(q^2)\simeq -\frac{\alpha}{2\pi}\int _0^1 d\xi \left[\left(\frac{m^2-q^2/2}{m^2-q^2\xi(1-\xi)}-1 \right)\log\left(\frac{-q^2\ {\rm or}\ m^2}{\mu^2}\right)\right]\\
\simeq -\frac{\alpha}{2\pi}f_{\rm IR}(q^2)\log\left(\frac{-q^2\ {\rm or}\ m^2}{\mu^2}\right)\ \ (6.61)
\end{align}
And he define
\begin{align}
f_{\rm IR}(q^2)=\int _0^1 d\xi \left(\frac{m^2-q^2/2}{m^2-q^2\xi(1-\xi)}\right)-1\ \ (6.62)
\end{align}
then we can compute in $-q^2\to \infty$ limit,
\begin{align}
f_{\rm IR}(q^2)\to \log\left(\frac{-q^2}{m^2}\right)\ \ (6.64)
\end{align}
And he conclude that
\begin{align}
F_1(q^2)\simeq -\frac{\alpha}{2\pi}f_{\rm IR}(q^2)\log\left(\frac{-q^2\ {\rm or}\ m^2}{\mu^2}\right)=-\frac{\alpha}{2\pi}\log\left(\frac{-q^2}{m^2}\right)\log\left(\frac{-q^2}{\mu^2}\right)\ \ (6.65)
\end{align}
And he said this expression contains not only the correct coefficient of $\log(1/\mu^2)$, but also the correct coefficient of $\log^2 q^2$. I have several question;One- I don't understand why the numerators inside the log is not important when $\mu$ is small. Two- Why did you choose $-q^2$ for the last step, even though the contents of log could be either $-q^2$ or $m^2$? Three- I think this is a very bad approximation, but why does it reflect the correct coefficient of log?
 A: In response to the first part of your question: you can write $\log (X/\mu^2) = \log(X) - \log(\mu^2)$. In the limit $\mu\rightarrow 0$ for fixed $X$, the second term will dominate the first, for any $X$.
For the second part of the question, I think this is driven by physics. There's no mathematical reason to choose $-q^2$ or $m^2$ to appear in the log, since we've established that the numerator doesn't matter. However, on physical grounds, we expect renormalization to tell us how physical quantities depend on the energy scale, $q^2$. So we are looking to establish how the answer depends on $\log(-q^2/\mu^2)$. It would be correct, but quite boring (since $m^2$ is a constant), to write $\log(m^2/\mu^2)$.
I think perhaps the trick is that this result was derived in the limit $\mu\rightarrow 0$ (in which case any numerator is equivalent), but the final result essentially includes a correction to this limit (since we are trusting the appearance of $q^2$ in the numerator of the log). I think that the difference between different prescriptions for what to put in the numerator of the log can be absorbed in the definition of how $\alpha$ depends on $\mu$ in the renormalization group flow.
Perhaps there is also a more careful but more difficult way to take the limit, which gives the same answer, and Peskin and Schroeder are reverse engineering a simpler argument to get to the result faster. I don't know this to be the case, however.
