I'm trying to solve the equation
$$ \mathbb{E}_X[\log (a + X)] = b$$
to $a$ where $b>0$ and $X$ is a positive random variable distributed according to $P(X)$. The solution can be written as a function of the moments of $X$, or any other statistic of $X$.
Is this possible? I'm struggling to understand what transformation would even help.
In my case $P(X) = (\alpha - 1) X^{-\alpha}$, $\alpha > 2$, such that $\mathbb{E}_X[\log X] = 1/(\alpha - 1)$.