Given a random variable $X\sim {\text Gamma} (\alpha ,\beta )$, and define $Y=\ln(X)\cdot \mathbb{I}\{X\le a\}$, does it exist a closed-form expression for $\mathbb{E}\{Y\}$ or an approximated one?
In detail, $$\mathbb{E}\{Y\}=\int_0^a\ln(X)\,dF_X(x)$$ where $F_X(x)=\frac {\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}$.
If we know the CDF of $Y$, we can solve it using a similar method in here.