I am reading about the value of p-adic L-function $L_p(\theta,s)$ at $s=1$. Someone claims the following formula: \begin{equation} L_p(\theta,1):=\int_{\mathbb{Z}_p^{\times}}x^{-1} \cdot \mu_{\theta}=\mathcal{A}_{Res_{\mathbb{Z}_{p}^{\times}}}(x^{-1}\mu_\theta)(0) \end{equation} The Last equation means the Mahler transform over $\mathbb{Z}_p$ and restricted to $\mathbb{Z}_p^{\times}$
As to the formula, I have a question. How can the multiplication of $x^{-1}$ to a measure valid on $\mathbb{Z}_p$ since $x$ is not inversible on $\mathbb{Z}_p$. A more proper way to write the formula I suppose is $x^{-1}\mathcal{A}_{Res_{\mathbb{Z}_p^{\times}}(\mu_\theta)}(0)$. Is this right?
Furthermore, by calculating $x^{-1}\mathcal{A}_{Res_{\mathbb{Z}_p^{\times}}(\mu_\theta)}(0)$, I get the formula below: \begin{equation} x^{-1}\mathcal{A}_{Res_{\mathbb{Z}_p^{\times}}(\mu_\theta)}(T)=\frac{-1}{G(\theta)}\sum_{c\in(\mathbb{Z}/{N\mathbb{Z}})^{\times}}\theta(c)^{-1}log((1+T)\zeta^{c}-1)+\frac{\theta(p)p^{-1}}{G(\theta^{-1})}\sum_{c\in(\mathbb{Z}/{N\mathbb{Z}})^{\times}}\theta(c)^{-1}log((1+T)^{p}\zeta^{c}-1) \end{equation} ,where T is a variable and N is the conductor of $\theta$.
However, as we all known, the mahlor transform maps measures on $\mathbb{Z}_p$, especially on $\mathbb{Z}_p^{\times}$ as a subgroup of $\mathbb{Z}_p$, to the power series ring $\mathbb{Z}_p[[T]]$. So I suppose the formula above can be expanded as power series. But how to do this with the logaritm, consideing the loarithm can only be expressed as a power series only when it acts on $1+p\mathbb{Z}_p[T]$?