# Random variable with finite logarithmic first moment, infinite logarithmic variance [duplicate]

Could you provide an example of a random variable $$X$$ such that $$|\mathbb{E}(\ln(X))|<\infty$$ but $$\text{Var}(\ln(X))=\infty$$, if such a random variable exists at all?

• Let $Y$ be a positive variable with finite expectation and infinite variance, then set $X=\exp(Y).$
– whuber
Mar 4, 2021 at 18:31
• Richard, you should perhaps include the condition that $Var(X)\leq \infty$. Mar 5, 2021 at 0:36
• Sorry it should strict inequality above. Mar 5, 2021 at 4:25