I'm trying to calculate $\mathbb E[\log X]$ for X a non-negative random variable with finite mean, and
$$\mathbb E[\log X|X\leq t]=\frac{1}{F(t)}\int_0^t \log xf(x)dx=\frac{1}{F(t)}([\log zF(z)]^t_0-\int^t_0\frac{1}{z}F(z)dz)$$ But now I don't know how to solve $[\log zF(z)]_{t=0}$ I believe this should be $0$ but have no ideal how to show it. Since $F$ is unknown L'Hôpital also doesn't work. Any hints would be appreciated!