Concentration of a random variable around its mean

Question

Consider a random variable $X$.

In general, is it true that the random variable $\log X$ is more tightly concentrated around its expected value than the random variable $X$?

Answer 1

No. Take $n\in\mathbb N$ and $Z\sim N(0,1)\,.$ Then $$ X:=e^{Z-1/2-n} $$ has \begin{align} \mathbb E[X]&=e^{-n}\,,\\[2mm] \operatorname{Var}[X]&=\mathbb E[e^{2Z-1-2n}]-e^{-2n}\\[2mm] &=e^{-2n+1}-e^{-2n}\\[2mm] &=e^{-2n}(e-1)\,, \end{align} which becomes arbitrarily small when $n$ is large. On the other hand $$ \operatorname{Var}[\log X]=\operatorname{Var}[Z]=1\,. $$