Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}\_p$. As in II.4.3 in [Colmez - Fonctions d'une variable $p$-adique](https://webusers.imj-prg.fr/~pierre.colmez/fonctionsdunevariable.pdf), I can restrict $\mu$ to $1+p\mathbf Z\_p$, and there is a formula which tells me how to compute the Amice transform in terms of the original one in a simple way: $$\mathcal{A}\_{\mathrm{Res}\_{1+p\mathbf{Z}\_p}(\mu)}(T) = \frac{1}{p} \sum\_{\eta \in \mu\_p} \eta^{-1} \mathcal{A}\_\mu((1+T)\eta-1)$$


But suppose I define a new measure $\tilde{\mu}$ on $\mathbf{Z}\_p$ by $$\int\_{\mathbf{Z}\_p} f(x) \cdot\tilde\mu(x) := \int\_{1+p\mathbf{Z}\_p}f(\log(x)/p)\cdot \mu(x)$$ which is by definition $$\int\_{\mathbf{Z}\_p} f(\log(x)/p) \mathbf{1}\_{1+p\mathbf{Z}\_p}(x)\cdot\mu(x)$$ and equal to $$\int\_{\mathbf{Z}\_p} f(\log(x)/p) \cdot \mathrm{Res}\_{1+p\mathbf{Z}\_p}(\mu)(x)$$
Then is there any way to get my hands on the Amice transform $\mathcal{A}\_{\tilde{\mu}}$ in terms of $\mathcal{A}\_\mu$, as above?



Change of variables for $p$-adic integral
If I understand correctly, the answer is probably not, since it would imply that the Iwasawa power series of a $p$-adic $L$-function has a "simple" relation with the original measure on $1+p\mathbb{Z}\_p$. But the Iwasawa power series is usually hard to compute (explicitly). But, there are relations between your two measures, for example, under certain assumptions, the mu and lambda invariants of both measures are related. You may take a look at the following articles:


[Warren Sinnott, On the $\mu$-invariant of the $\Gamma$-transform of a rational function](https://www.digizeitschriften.de/dms/toc/?PID=PPN356556735_0075)


[Yûji Kida, The $\lambda$-invariants of $p$-adic measures on ${\bf Z}\_p$ and $1+q{\bf Z}\_p$](https://kanazawa-u.repo.nii.ac.jp/?action=pages_view_main&active_action=repository_view_main_item_detail&item_id=11211&item_no=1&page_id=13&block_id=21)


[Nancy Childress, $\lambda$-invariants and $\Gamma$-transforms](https://www.digizeitschriften.de/dms/toc/?PID=PPN365956996_0064)


[Junya Satoh, Iwasawa $\lambda$-invariants of $\Gamma$-transforms](https://www.sciencedirect.com/science/article/pii/0022314X92900865)


(Your question is from July, so this comment is late. It would be interesting to know how you solved this problem.)


