I am looking at a proof of the Central Limit Theorem and I am having some trouble understanding the following step:
We have the Characteristic funtion of a distribution $w(x)$: \begin{equation} \xi(k) = \sum_{n=0}^{\infty} \frac{(-ik)^n}{n!} \langle X^n \rangle \end{equation}
at some point in the proof the author says that he takes the exponential of the $ln$ of this function, and skipst the actual computation leaving me with: \begin{equation} \xi(k) = \exp[-ik \langle X \rangle -\frac{1}{2}k^2(\Delta x)^2 + ...k^3 + ...] \end{equation}
I am struggling to comprend what exactly has been done here,since i do not seem to work it appropriately with logarithmus prorpieties. Can somebody help?