Can Elog(x) Be Infinite for Some Distributions?

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For a random variable x > 0 with a finite mean, Jensen's inequality indicates that E(log(x)) is less than or equal to log(E(x)), which is finite. The discussion raises the question of whether E(log(x)) can be negative infinity, particularly as x approaches zero, where log(x) tends to negative infinity. An example is provided using the distribution x = exp(-1/u) with u uniformly distributed on (0,1). The consensus is that while log(x) is finite for x > 0, the behavior of E(log(x)) can lead to negative infinity under certain conditions. Therefore, E(log(x)) cannot be infinite if x is strictly positive.
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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
 
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If x> 0 then log(x) is always finite and so E(log(x)) must be finite.
 
But when x -> 0 , log(x) -> -inf
 
Try x=exp(-1/u) where u is uniform on (0,1).
 

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