I have two functions: $$\log P=e^{-2\pi\xi_{o}^{2}t}\frac{t}{\tau}$$ and $$\log P=\left(E_{0}^{2}\tau t\right)^{-2}\frac{t}{\tau}$$ And they touch at some point, there's a transition of the first function to the second. I want to know the time of transition, so I equalized the two functions and obtained: $$e^{-2\pi\xi_{o}^{2}t}=\left(E_{0}^{2}\tau t\right)^{-2}$$ $$-2\pi\xi_{o}^{2}t=\log\left(E_{0}^{2}\tau t\right)^{-2}$$ $$-2\pi\xi_{o}^{2}t=-2\log\left(E_{0}^{2}\tau t\right)$$ $$\pi\xi_{o}^{2}t=\log\left(E_{0}^{2}\tau t\right)$$ I get to this. But i was supposed to get to the time of transition of: $$t\approx\tau log(E_{0}\tau)$$ I don't understand what i am doing wrong.
These are functions i obtained from a decayment graph. $ξ_{0}$ is the coupling coefficient between a discrete state and a continuum of states, $E_{0}$ is the energy of the discrete state and $τ$ is the decayment time, where $τ=1/2π|ξ_{E}|^2$.
