The Beta Function and the Bare Charge According to "Introductory Lectures on Quantum Field Theory", by L. Álvarez-Gaumé and M. A. Vázquez-Mozo, (P. 84) the dependence of the bare charge $e_{0}(\Lambda)$ on the cutoff $\Lambda$ is determined by the identity
$$ e(\mu)^{2} = e_{0}(\Lambda)^{2}\left( 1+\frac{e_{0}(\Lambda)^{2}}{12\pi^{2}}\log \left( \frac{\mu^{2}}{\Lambda^{2}}\right)\right).\tag{8.13}$$
Taking into account that we are working in perturbation theory in $e(\mu)^{2}$, we can express the bare charge $e_{0}(\Lambda)^{2}$ in terms of $e(\mu)^{2}$ as
$$e(\Lambda)^{2} = e(\mu)^{2}\left( 1+\frac{e(\mu)^{2}}{12\pi^{2}}\log \left( \frac{\mu^{2}}{\Lambda^{2}}\right)\right) + O(e(\mu)^{
6}).\tag{8.14}$$
How does one obtain the second equation? I tried solving for the bare charge in the first one and then replacing it twice in the result, but I end up with an extra minus sign.
 A: Your second equation has the sign wrong (obviously; read on). It is probably a typo. It should be, instead,
$$e(\Lambda)^{2} = e(\mu)^{2}\left( 1+\frac{e(\mu)^{2}}{12\pi^{2}}\log \left( \frac{\Lambda^{2}}{\mu^{2}}\right)\right) + O(e(\mu)^{
6}),\tag{8.14}$$
which results from inverting
$$ e(\mu)^{2} = e (\Lambda)^{2}\left( 1+\frac{e (\Lambda)^{2}}{12\pi^{2}}\log \left( \frac{\mu^{2}}{\Lambda^{2}}\right)\right),\tag{8.13}$$
since
$$
{ e(\Lambda)^{2}\over e(\mu)^{2}}= { 1\over 1+\frac{e (\Lambda)^{2}}{12\pi^{2}}\log \left( \frac{\mu^{2}}{\Lambda^{2}}\right )  }\\ =  1-\frac{e (\Lambda)^{2}}{12\pi^{2}}\log \left( \frac{\mu^{2}}{\Lambda^{2}}\right ) +... \\ =  1+\frac{e (\mu)^{2}}{12\pi^{2}}\log \left( \frac{\Lambda^{2}}{\mu^{2}}\right ) +...
$$
It is this "self-similar scale" feature (the term with 12 in the denominator  agreement with the denominator of the log argument) that makes elimination of $\log \Lambda^2$ possible to yield the epochal 1954 Gell-Mann—Low RG equation,
$${e(\mu)^{2}\over e(\mu_0 )^{2} } =  1+\frac{e(\mu_0)^{2}}{12\pi^{2}}\log \left( \frac{\mu^{2}}{\mu_0^{2}}\right)  .\tag{8.15}$$
$e(\mu)$ is an increasing function of μ.
(Appendix B, with suitable gratitude expressed to T D Lee by the authors: a major moment in 20th century intellectual history. Feynman has said it weighed heavily in his hiring MGM at Caltech.
Geeky: In WP, consider $~  G (x) =\exp( 1/x)$ as your Wegner scaling function and $d= -1/6\pi^2$.)
