Characteristic function of logarithm of random variable

If I know the characteristic function ${\varphi}_{X}(t)$ of a random variable $X>0$, how can I write the characteristic function ${\varphi}_{Y}(t)$ of $Y=\mathrm{log}(X)$?

I know that ${\varphi}_{X}(t)=E[{e}^{itX}]$ and ${\varphi}_{Y}(t)=E[{e}^{it\mathrm{log}(X)}]$. But I can't derive one from the other. Any idea? I would like to use ${\varphi}_{X}(t)$ to calculate the second moments $X$

If I know the characteristic function ${\varphi}_{X}(t)$ of a random variable $X>0$, how can I write the characteristic function ${\varphi}_{Y}(t)$ of $Y=\mathrm{log}(X)$?

I know that ${\varphi}_{X}(t)=E[{e}^{itX}]$ and ${\varphi}_{Y}(t)=E[{e}^{it\mathrm{log}(X)}]$. But I can't derive one from the other. Any idea? I would like to use ${\varphi}_{X}(t)$ to calculate the second moments $X$