Show that $\psi_{X}^{k}\left(1\right)=E\left[X\left(X-1\right)\left(X-2\right)\cdots\left(X-k-1\right)\right]$

Question

Show that $$\psi_{X}^{k}\left(1\right)=E\left[X\left(X-1\right)\left(X-2\right)\cdots\left(X-k-1\right)\right]$$ and hence that $\psi_{X}\left(t\right)$ generates factorial moments. Assume $X$ is continuous. I'm stuck in this step: Let $$M_{X}\left(logt\right)=E\left[e^{log\left(t\right)X}\right]=\int_{-\infty}^{+\infty}e^{log\left(t\right)x}f\left(x\right)dx$$ According to the text book, it can be use the expansion $$e^{z}=1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$$ but not sure if this is the right way to show what is asked. Any help would be appreciated