Can Elog(x) Be Infinite for Some Distributions?

[St41n]

Let x>0 be a random variable with some distribution with finite mean and let E denote the expectation with respect to that distribution.
By Jensen's inequality we have Elog(x) =< logE(x) < +inf

But, does this imply that -inf < Elog(x) too? Or is it possible that Elog(x) = -inf

Sorry if my question is stupid. Thx in advance

 

[HallsofIvy]

If x> 0 then log(x) is always finite and so E(log(x)) must be finite.

 

[St41n]

But when x -> 0 , log(x) -> -inf

 

[bpet]

Try x=exp(-1/u) where u is uniform on (0,1).